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Chapter 1

Abstract logic

For other uses of “Abstract logic”, see Abstract logic (disambiguation).

In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relationwith specific properties related to occurrence, expansion, isomorphism, renaming and quantification.[1]

Based on Lindström's characterization, first order logic is, up to equivalence, the only abstract logic which is countablycompact and has Löwenheim number ω.[2]

1.1 See also• Abstract algebraic logic

• Abstract model theory

• Löwenheim number

• Lindström’s theorem

• Universal logic

1.2 References[1] C. C. Chang and Jerome Keisler Model Theory, 1990 ISBN 0-444-88054-2 page 128

[2] C. C. Chang and Jerome Keisler Model Theory, 1990 ISBN 0-444-88054-2 page 132

2

Chapter 2

Abstract model theory

In mathematical logic, abstract model theory is a generalization of model theory which studies the general propertiesof extensions of first-order logic and their models.[1]

Abstract model theory provides an approach that allows us to step back and study a wide range of logics and theirrelationships.[2] The starting point for the study of abstract models, which resulted in good examples was Lindström’stheorem.[3]

In 1974 Jon Barwise provided an axiomatization of abstract model theory.[4]

2.1 See also• Lindström’s theorem

• Institution (computer science)

• Institutional model theory

2.2 Notes[1] Institution-independent model theory by Răzvan Diaconescu 2008 ISBN 3-7643-8707-6 page 3

[2] Handbook of mathematical logic by Jon Barwise 1989 ISBN 0-444-86388-5 page 45

[3] Jean-Yves Béziau Logica universalis: towards a general theory of logic 2005 ISBN 978-3-7643-7259-0 pages 20–25

[4] J. Barwise, 1974 Axioms for abstract model theory, Annals of Math. Logic 7:221–265

2.3 Further reading• Jon Barwise; Solomon Feferman (1985). Model-theoretic logics. Springer-Verlag. ISBN 978-0-387-90936-3.

3

Chapter 3

Abstract structure

An abstract structure in mathematics is a formal object that is defined by a set of laws, properties, and relationshipsin a way that is logically if not always historically independent of the structure of contingent experiences, for example,those involving physical objects. Abstract structures are studied not only in logic and mathematics but in the fieldsthat apply them, as computer science, and in the studies that reflect on them, such as philosophy and especially thephilosophy of mathematics. Indeed, modern mathematics has been defined in a very general sense as the study ofabstract structures (by the Bourbaki group: see discussion there, at algebraic structure and also structure).An abstract structure may be represented (perhaps with some degree of approximation) by one or more physicalobjects — this is called an implementation or instantiation of the abstract structure. But the abstract structure itselfis defined in a way that is not dependent on the properties of any particular implementation.An abstract structure has a richer structure than a concept or an idea. An abstract structure must include preciserules of behaviour which can be used to determine whether a candidate implementation actually matches the abstractstructure in question. Thus we may debate how well a particular government fits the concept of democracy, but thereis no room for debate over whether a given sequence of moves is or is not a valid game of chess.

3.1 Examples

A sorting algorithm is an abstract structure, but a recipe is not, because it depends on the properties and quantities ofits ingredients.A simple melody is an abstract structure, but an orchestration is not, because it depends on the properties of particularinstruments.Euclidean geometry is an abstract structure, but the theory of continental drift is not, because it depends on thegeology of the Earth.A formal language is an abstract structure, but a natural language is not, because its rules of grammar and syntax areopen to debate and interpretation.

3.2 See also• Abstraction in computer science• Abstraction in general• Abstraction in mathematics• Abstract object• Deductive apparatus• Formal sciences• Mathematical structure

4

Chapter 4

Aczel’s anti-foundation axiom

In the foundations of mathematics,Aczel’s anti-foundation axiom is an axiom set forth by Peter Aczel (1988), as analternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directedgraph corresponds to a unique set. In particular, according to this axiom, the graph consisting of a single vertex witha loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom isnecessarily a non-well-founded set theory.

4.1 Accessible pointed graphs

An accessible pointed graph is a directed graph with a distinguished vertex (the “root”) such that for any node in thegraph there is at least one path in the directed graph from the root to that node.The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of aunique set. For example, the directed graph with only one node and an edge from that node to itself corresponds toa set of the form x = x.

4.2 See also• von Neumann universe

4.3 References• Aczel, Peter (1988). Non-well-founded sets. (PDF). CSLI Lecture Notes 14. Stanford, CA: Stanford Univer-sity, Center for the Study of Language and Information. ISBN 0-937073-22-9. MR 0940014.

• Goertzel, Ben (1994). “Self-Generating Systems”. Chaotic Logic: Language, Thought and Reality From thePerspective of Complex Systems Science. Plenum Press. ISBN 978-0-306-44690-0. Retrieved 2007-01-15.

• Akman, Varol; Pakkan, Mujdat (1996). “Nonstandard set theories and information management” (PDF).Journal of Intelligent Information Systems 6 (1): 5–31. doi:10.1007/BF00712384.

5

Chapter 5

AD+

In set theory,AD+ is an extension, proposed byW. HughWoodin, to the axiom of determinacy. The axiom, which isto be understood in the context of ZF plus DCR (the axiom of dependent choice for real numbers), states two things:

1. Every set of reals is ∞-Borel.

2. For any ordinal λ less than Θ, any subset A of ωω, and any continuous function π:λω→ωω, the preimage π−1[A]is determined. (Here λω is to be given the product topology, starting with the discrete topology on λ.)

The second clause by itself is referred to as ordinal determinacy.

5.1 See also• Suslin’s problem

5.2 References• W.H. Woodin The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (1999 Walter deGruyter) p. 618

6

Chapter 6

Adequate pointclass

In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursivepointsets and is closed under recursive substitution, bounded universal and existential quantification and preimagesby recursive functions.[1][2]

6.1 References[1] Moschovakis, Y. N. (1987), Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics, Elsevier, p. 158,

ISBN 9780080963198.

[2] Gabbay, Dov M.; Kanamori, Akihiro; Woods, John (2012), Sets and Extensions in the Twentieth Century, Handbook of theHistory of Logic 6, Elsevier, p. 465, ISBN 9780080930664.

7

Chapter 7

Admissible ordinal

In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model ofKripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα⊧Σ0-collection.[1][2]

The first two admissible ordinals are ω and ωCK1 (the least non-recursive ordinal, also called the Church–Kleene

ordinal).[2] Any regular uncountable cardinal is an admissible ordinal.By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to theChurch-Kleene ordinal, but for Turing machines with oracles.[1] One sometimes writes ωCK

α for the α -th ordinalwhich is either admissible or a limit of admissibles; an ordinal which is both is called recursively inaccessible.[3] Thereexists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can definerecursively Mahlo cardinals, for example).[4] But all these ordinals are still countable. Therefore, admissible ordinalsseem to be the recursive analogue of regular cardinal numbers.Notice that α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ<α for which thereis a Σ1(Lα) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissibleordinal.

7.1 See also• Large countable ordinals

• Inaccessible cardinal

• Constructible universe

7.2 References[1] Friedman, Sy D. (1985), “Fine structure theory and its applications”, Recursion theory (Ithaca, N.Y., 1982), Proc. Sympos.

Pure Math. 42, Amer. Math. Soc., Providence, RI, pp. 259–269, doi:10.1090/pspum/042/791062, MR 791062. See inparticular p. 265.

[2] Fitting, Melvin (1981), Fundamentals of generalized recursion theory, Studies in Logic and the Foundations ofMathematics105, North-Holland Publishing Co., Amsterdam-New York, p. 238, ISBN 0-444-86171-8, MR 644315.

[3] Friedman, Sy D. (2010), “Constructibility and class forcing”, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht,pp. 557–604, doi:10.1007/978-1-4020-5764-9_9, MR 2768687. See in particular p. 560.

[4] Kahle, Reinhard; Setzer, Anton (2010), “An extended predicative definition of the Mahlo universe”,Ways of proof theory,Ontos Math. Log. 2, Ontos Verlag, Heusenstamm, pp. 315–340, MR 2883363.

8

Chapter 8

Admissible set

In set theory, a discipline within mathematics, an admissible set is a transitive set A such that ⟨A,∈⟩ is a model ofKripke–Platek set theory (Barwise 1975).The smallest example of an admissible set is the set of hereditarily finite sets. Another example is the set of hereditarilycountable sets.

8.1 See also• Admissible ordinal

8.2 References• Barwise, Jon (1975). Admissible Sets and Structures: An Approach to Definability Theory, Perspectives inMathematical Logic, Volume 7, Springer-Verlag. Electronic version on Project Euclid.

9

Chapter 9

Algebraic definition

In mathematical logic, an algebraic definition is one that can be given using only equations between terms with freevariables. Inequalities and quantifiers are specifically disallowed.Saying that a definition is algebraic is a stronger condition than saying it is elementary.

9.1 Related• Algebraic sentence

• Algebraic theory

• Algebraic expression.

• Algebraic equation.

10

Chapter 10

Algebraic semantics (mathematical logic)

In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraiclogic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, booleanalgebras with an interior operator. Other modal logics are characterized by various other algebras with operators. Theclass of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositionalintuitionistic logic.

10.1 Further reading• Josep Maria Font; Ramón Jansana (1996). A general algebraic semantics for sentential logics. Springer-Verlag.ISBN 9783540616993. (2nd published by ASL in 2009) open access at Project Euclid

• W.J. Blok; Don Pigozzi (1989). Algebraizable logics. American Mathematical Society. ISBN 0821824597.

• Janusz Czelakowski (2001). Protoalgebraic logics. Springer. ISBN 9780792369400.

• J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford UniversityPress. ISBN 9780198531920. Good introduction for readers with prior exposure to non-classical logics butwithout much background in order theory and/or universal algebra; the book covers these prerequisites atlength. The book however has been criticized for poor and sometimes incorrect presentation of AAL results.

11

Chapter 11

Algebraic sentence

In mathematical logic, an algebraic sentence is one that can be stated using only equations between terms with freevariables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logicinvolving only algebraic sentences.Saying that a sentence is algebraic is a stronger condition than saying it is elementary.

11.1 Related• Algebraic theory

• Algebraic definition

• Algebraic expression

12

Chapter 12

Algorithmic logic

Algorithmic logic is a calculus of programs which allows the expression of semantical properties of programs byappropriate logical formulas. It provides a framework that enables proving the formulas from the axioms of programconstructs such as assignment, iteration and composition instructions and from the axioms of the data structures inquestion see Mirkowska & Salwicki (1987), Banachowski et al. (1977).

The following diagram helps to locate algorithmic logic among other logics.

Propositional logicorSentential calculus

⊂

Predicate calculusorFirst order logic

⊂ Calculus of programsorAlgorithmic logic

The formalized language of algorithmic logic (and of algorithmic theories of various data structures) contains threetypes of well formed expressions: terms - i.e. expressions denoting operations on elements of data structures, for-mulas - i.e. expressions denoting the relations among elements of data structures, programs - i.e. algorithms - theseexpressions describe the computations. For semantics of terms and formulas consult pages on first order logic andTarski’s semantic. The meaning of a programK is the set of possible computations of the program.Algorithmic logic is one of many logics of programs. Another logic of programs is dynamic logic, see dynamic logic,Harel, Kozen & Tiuryn (2000).

12.1 Footnotes

12.2 Bibliography1. [Mirkowska & Salwicki] |Mirkowska, Grażyna; Salwicki, Andrzej (1987). Algorithmic Logic. Warszawa &

Boston: PWN & D. Reidel Publ. p. 372. ISBN 8301068590.

2. [Banachowski et al.] |Banachowski, Lech; Kreczmar, Antoni; Mirkowska, Grażyna; Rasiowa, Helena; Sal-wicki, Andrzej (1977). An introduction to Algorithmic Logic - Metamathematical Investigations of Theory ofPrograms. Banach Center Publications 2. Warszawa: PWN. pp. 7–99.

3. Harel, David; Kozen, Dexter; Tiuryn, Jerzy (2000). Dynamic Logic. Cambridge Massachusetts: MIT Press.p. 459.

13

Chapter 13

Aronszajn line

In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinalityℵ1 which contains no subset order-isomorphic to

• ω1 with the usual ordering

• the reverse of ω1

• an uncountable subset of the Real numbers with the usual ordering.

Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linearordering is an Aronszajn line if and only if it is the lexicographical ordering of some Aronszajn tree.[1]

13.1 References[1] Funk, Will; Lutzer, David J. (2005). “Lexicographically ordered trees”. Topology and its Applications 152 (3): 275–300.

doi:10.1016/j.topol.2004.10.011. Zbl 1071.03032.

14

Chapter 14

Automated proof checking

Automated proof checking is the process of using software for checking proofs for correctness. It is one of themost developed fields in automated reasoning.Automated proof checking differs from automated theorem proving in that automated proof checking simply me-chanically checks the formal workings of an existing proof, instead of trying to develop new proofs or theorems itself.Because of this, the task of automated proof verification is much simpler than that of automated theorem proving,allowing automated proof checking software to be much simpler than automated theorem proving software.Because of this small size, some automated proof checking systems can have less than a thousand lines of core code,and are thus themselves amenable to both hand-checking and automated software verification.The Mizar system, HOL Light, and Metamath are examples of automated proof checking systems.Automated proof checking can be done either as a batch operation, or interactively, as part of an interactive theoremproving system.

14.1 Field journals and conferences

• Intelligent Computer Mathematics

• Journal of Formalized Reasoning

• Interactive Theorem Proving

• Formalized Mathematics

• Studies in Logic, Grammar and Rhetoric

14.2 See also

• Computer-aided proof

• Formal verification

• Proof assistant

• QED manifesto

14.3 External links

• Julie Rehmeyer (November 14, 2008). “How to (really) trust a mathematical proof”. ScienceNews. Retrieved2008-11-14.

15

16 CHAPTER 14. AUTOMATED PROOF CHECKING

• Metamath: a proof checking system with an extensive collection of machine-readable proofs covering a con-siderable range of mathematical fields

• Digimath: Freek Wiedijk’s alphabetic list of systems

• MathSystem: Mathematical Software systems

Chapter 15

Axiom of adjunction

In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ y givenby “adjoining” the set y to the set x.

∀x ∀y ∃w ∀z [z ∈ w ↔ (z ∈ x ∨ z = y)].

Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of settheory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as generalset theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursiveset functions.Tarski and Smielew showed that Robinson arithmetic can be interpreted in a weak set theory whose axioms areextensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34).

15.1 References• Bernays, Paul (1937), “A System ofAxiomatic Set Theory--Part I”, The Journal of Symbolic Logic (Associationfor Symbolic Logic) 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862

• Kirby, Laurence (2009), “Finitary Set Theory”, Notre Dame J. Formal Logic 50 (3): 227–244, MR 2572972

• Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amster-dam: North-Holland Publishing Company, MR 0058532

• Tarski, A., and Givant, Steven (1987) A Formalization of Set Theory without Variables. Providence RI: AMSColloquium Publications, v. 41.

17

Chapter 16

Axiom of projective determinacy

In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only toprojective sets.The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect informationof length ω in which the players play natural numbers, if the victory set (for either player, since the projective setsare closed under complementation) is projective, then one player or the other has a winning strategy.The axiom is not a theorem of ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD),which contradicts the axiom of choice, it is not known to be inconsistent with ZFC. PD follows from certain largecardinal axioms, such as the existence of infinitely many Woodin cardinals.PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfectset property and the property of Baire. It also implies that every projective binary relation may be uniformized by aprojective set.

16.1 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-

ican Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913. JSTOR1990913.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

18

Chapter 17

Axiom of real determinacy

In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states thefollowing:

Consider infinite two-person games with perfect information. Then, every game of length ω where bothplayers choose real numbers is determined, i.e., one of the two players has a winning strategy.

The axiom of real determinacy is a stronger version of the axiom of determinacy, which makes the same statementabout games where both players choose integers; it is inconsistent with the axiom of choice. ADR also implies theexistence of inner models with certain large cardinals.ADR is equivalent to AD plus the axiom of uniformization.

19

Chapter 18

Axiom schema of predicative separation

In axiomatic set theory, the axiom schema of predicative separation, or of restricted, or Δ0 separation, is aschema of axioms which is a restriction of the usual axiom schema of separation in Zermelo–Fraenkel set theory. Itonly asserts the existence of a subset of a set if that subset can be defined without reference to the entire universeof sets. The axiom appears in the systems of constructive set theory CST and CZF, as well as in the system ofKripke–Platek set theory. The name Δ0 comes from the Levy hierarchy (in analogy with the arithmetic hierarchy).The formal statement of this is the same as full separation schema, but with a restriction on the formulas that may beused. For any formula φ:

∀x ∃y ∀z (z ∈ y ↔ z ∈ x ∧ ϕ(z))

provided, as usual, that the variable y is not free in φ; but also provided that φ contains only bounded quantifiers. Thatis, all quantifiers in φ (if there are any) must appear in the form ∃x ∈ y ψ(x) or ∀x ∈ y ψ(x) for some sub-formulaψ.The meaning of this is that, given any set x, and any predicate φ there is a set y whose elements are the elements ofx which satisfy φ, provided φ only quantifies over existing sets, and never quantifies over all sets. This restriction isnecessary from a predicative point of view, since the universe of all sets contains the set being defined. If it werereferenced in the definition of the set, the definition would be circular.Although the schema contains one axiom for each restricted formula φ, it is possible in CZF to replace this schemawith a finite number of axioms.

20

Chapter 19

Barwise compactness theorem

In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usualcompactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwisein 1967.

19.1 Statement of the theorem

Let A be a countable admissible set. Let L be an A -finite relational language. Suppose Γ is a set of LA -sentences,where Γ is a Σ1 set with parameters from A , and every A -finite subset of Γ is satisfiable. Then Γ is satisfiable.

19.2 References• Barwise, J. (1967). Infinitary Logic and Admissible Sets (Ph. D. Thesis). Stanford University.

• C. J. Ash; Knight, J. (2000). Computable Structures and the Hyperarithmetic Hierarchy. Elsevier. p. 366.ISBN 0-444-50072-3.

• Jon Barwise; Solomon Feferman; John T. Baldwin (1985). Model-theoretic logics. Springer-Verlag. p. 295.ISBN 3-540-90936-2.

19.3 External links• Stanford Encyclopedia of Philosophy, “Infinitary Logic”, Section 5, “Sublanguages of L(ω1,ω) and the BarwiseCompactness Theorem”

21

Chapter 20

Bernays–Schönfinkel class

The Bernays–Schönfinkel class (also known as Bernays–Schönfinkel-Ramsey class) of formulas, named afterPaul Bernays and Moses Schönfinkel (and Frank P. Ramsey), is a decidable fragment of first-order logic formulas.It is the set of satisfiable formulas which, when written in prenex normal form, have an ∃∗∀∗ quantifier prefix and donot contain any function symbols.This class of logic formulas is also sometimes referred as effectively propositional (EPR) since it can be effectivelytranslated into propositional logic formulas by a process of grounding or instantiation.The decision problem for this class is NEXPTIME-complete.[1]

20.1 See also• Prenex normal form

20.2 References[1] Harry R. Lewis, Complexity Results for Classes of Quantificational Formulas, J. Computer and System Sciences, 21, 317-

353 (1980) doi:10.1016/0022-0000(80)90027-6

• Ramsey, F. (1930), “On a problem in formal logic”, Proc. LondonMath. Soc. 30: 264–286, doi:10.1112/plms/s2-30.1.264

• Piskac, R.; deMoura, L.; Bjorner, N. (December 2008), “Deciding Effectively Propositional Logic with Equal-ity” (PDF), Microsoft Research Technical Report (2008-181)

22

Chapter 21

Beth definability

In mathematical logic, Beth definability is a result that connects implicit definability of a property to its explicitdefinability, specifically the theorem states that the two senses of definability are equivalent.

21.1 Statement

The theorem states that, given any two models A and B of a first-order theory T in the language L' ⊇ L such that A|L= B|L (where A|L is the reduct of A to L), it is the case that A ⊨ φ[a] if and only if B ⊨ φ[a] (for φ a formula in L'and for all tuples a of A) only if it is also the case that φ is equivalent modulo T to a formula ψ in L. Less formally: aproperty is implicitly definable in a theory in language L (via introduction of a new symbol φ of an extended languageL') only if that property is explicitly definable in that theory (by formula ψ in the original language L).Clearly the converse holds as well, so that we have an equivalence between implicit and explicit definability. That is,a “property” is implicitly definable with respect to a theory if and only if it is explicitly definable.The theorem does not hold if the condition is restricted to finite models. We may have A ⊨ φ[a] if and only if B ⊨φ[a] for all pairs A,B of finite models without there being any L-formula ψ equivalent to φ modulo T.The result was first proven by Evert Willem Beth.

21.2 Sources• Hodges W. A Shorter Model Theory. Cambridge University Press, 1997.

23

Chapter 22

Binary decision

A binary decision is a choice between two alternatives, for instance between taking some specific action or not takingit.[1]

Binary decisions are basic to many fields. Examples include:

• Truth values in mathematical logic, and the corresponding Boolean data type in computer science, representinga value which may be chosen to be either true or false.[2]

• Conditional statements (if-then or if-then-else) in computer science, binary decisions about which piece ofcode to execute next.[3]

• Decision trees and binary decision diagrams, representations for sequences of binary decisions.[4]

• Binary choice, a statistical model for the outcome of a binary decision.[5]

22.1 References[1] Snow, Roberta M.; Phillips, Paul H. (2007),Making Critical Decisions: A Practical Guide for Nonprofit Organizations, John

Wiley & Sons, p. 44, ISBN 9780470185032.

[2] Dixit, J. B. (2009), Computer Fundamentals and Programming in C, Firewall Media, p. 61, ISBN 9788170088820.

[3] Yourdon, Edward (March 19, 1975), “Clear thinking vital: Nested IFs not evil plot leading to program bugs”,Computerworld:15.

[4] Clarke, E. M.; Grumberg, Orna; Peled, Doron (1999), Model Checking, MIT Press, p. 51, ISBN 9780262032704.

[5] Ben-Akiva, Moshe E.; Lerman, Steven R. (1985), Discrete Choice Analysis: Theory and Application to Travel Demand,Transportation Studies 9, MIT Press, p. 59, ISBN 9780262022170.

24

Chapter 23

Blake canonical form

In Boolean logic, a formula for a Boolean function f is in Blake canonical form, also called the complete sum ofprime implicants,[1] the complete sum,[2] or the disjunctive prime form,[3] when it is a disjunction of all the primeimplicants of f.[4] Blake canonical form is a disjunctive normal form.The Blake canonical form is not necessarily minimal, however all the terms of a minimal sum are contained in theBlake canonical form.[2]

It was introduced in 1937 by Archie Blake, who called it the “simplified canonical form";[5] it was named in honorof Blake by Frank Markham Brown in 1990.[4]

Blake discussed three methods for calculating the canonical form: exhaustion of implicants, iterated consensus, andmultiplication. The iterated consensus method was rediscovered by Samson and Mills, Quine, and Bing.[4]

23.1 See also• Horn clause

23.2 Notes[1] Tsutomu Sasao, “Ternary Decision Diagrams and their Applications”, in Tsutomu Sasao, Masahira Fujita, eds., Represen-

tations of Discrete Functions ISBN 0792397207, 1996, p. 278

[2] Abraham Kandel, Foundations of Digital Logic Design, p. 177

[3] Donald E. Knuth, The Art of Computer Programming 4A: Combinatorial Algorithms, Part 1, 2011, p. 54

[4] Frank Markham Brown, “The Blake Canonical Form”, chapter 4 of Boolean Reasoning: The Logic of Boolean Equations,ISBN 0486427854, 2nd edition, 2012, p. 77ff (first edition, 1990)

[5] “Canonical expressions in Boolean algebra”, Dissertation, Dept. of Mathematics, U. of Chicago, 1937, reviewed in J. C.C. McKinsey, The Journal of Symbolic Logic 3:2:93 (June 1938) doi:10.2307/2267634 JSTOR 2267634

25

Chapter 24

Boolean domain

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpre-tations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usuallywritten as 0, 1,[1][2][3] false, true, F, T,[4] ⊥,⊤ [5] or B. [6][7]

The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. Theinitial object in the category of bounded lattices is a Boolean domain.In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programminglanguages feature reserved words or symbols for the elements of the Boolean domain, for example false and true.However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, forexample, falsity is represented by the number 0 and truth is represented by the number 1 or −1 respectively, and allvariables that can take these values can also take any other numerical values.

24.1 Generalizations

The Boolean domain 0, 1 can be replaced by the unit interval [0,1], in which case rather than only taking values 0or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1−x,conjunction (AND) is replaced with multiplication ( xy ), and disjunction (OR) is defined via De Morgan’s law to be1− (1− x)(1− y) .Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logicand probabilistic logic. In these interpretations, a value is interpreted as the “degree” of truth – to what extent aproposition is true, or the probability that the proposition is true.

24.2 See also

• Boolean-valued function

24.3 Notes[1] Dirk van Dalen, Logic and Structure. Springer (2004), page 15.

[2] David Makinson, Sets, Logic and Maths for Computing. Springer (2008), page 13.

[3] George S. Boolos and Richard C. Jeffrey, Computability and Logic. Cambridge University Press (1980), page 99.

[4] Elliott Mendelson, Introduction to Mathematical Logic (4th. ed.). Chapman & Hall/CRC (1997), page 11.

[5] Eric C. R. Hehner, A Practical Theory of Programming. Springer (1993, 2010), page 3.

[6] Ian Parberry (1994). Circuit Complexity and Neural Networks. MIT Press. p. 65. ISBN 978-0-262-16148-0.

26

24.3. NOTES 27

[7] Jordi Cortadella; et al. (2002). Logic Synthesis for Asynchronous Controllers and Interfaces. Springer Science & BusinessMedia. p. 73. ISBN 978-3-540-43152-7.

Chapter 25

Boolean function

Not to be confused with Binary function.

In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ : Bk →B, where B = 0, 1 is a Boolean domain and k is a non-negative integer called the arity of the function. In the casewhere k = 0, the “function” is essentially a constant element of B.Every k-ary Boolean function can be expressed as a propositional formula in k variables x1, …, xk, and two propo-sitional formulas are logically equivalent if and only if they express the same Boolean function. There are 22k k-aryfunctions for every k.

25.1 Boolean functions in applications

A Boolean function describes how to determine a Boolean value output based on some logical calculation fromBoolean inputs. Such functions play a basic role in questions of complexity theory as well as the design of circuitsand chips for digital computers. The properties of Boolean functions play a critical role in cryptography, particularlyin the design of symmetric key algorithms (see substitution box).Boolean functions are often represented by sentences in propositional logic, and sometimes asmultivariate polynomialsover GF(2), but more efficient representations are binary decision diagrams (BDD), negation normal forms, andpropositional directed acyclic graphs (PDAG).In cooperative game theory, monotone Boolean functions are called simple games (voting games); this notion isapplied to solve problems in social choice theory.

25.2 See also

• Algebra of sets

• Boolean algebra

• Boolean algebra topics

• Boolean domain

• Boolean-valued function

• Logical connective

• Truth function

• Truth table

• Symmetric Boolean function

28

25.3. REFERENCES 29

• Decision tree model

• Evasive Boolean function

• Indicator function

• Balanced boolean function

• 3-ary Boolean functions

25.3 References• Crama, Y; Hammer, P. L. (2011), Boolean Functions, Cambridge University Press.

• Hazewinkel, Michiel, ed. (2001), “Boolean function”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Janković, Dragan; Stanković, Radomir S.; Moraga, Claudio (November 2003). “Arithmetic expressions opti-misation using dual polarity property” (PDF). Serbian Journal of Electrical Engineering 1 (71 - 80, number 1).Retrieved 2015-06-07.

• Mano, M. M.; Ciletti, M. D. (2013), Digital Design, Pearson.

Chapter 26

Borel equivalence relation

In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borelsubset of X × X (in the product topology).

26.1 Formal definition

Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducibleto F, in symbols E ≤B F, if and only if there is a Borel function

Θ : X → Y

such that for all x,x' ∈ X, one has

xEx' ⇔ Θ(x)FΘ(x' ).

Conceptually, if E is Borel reducible to F, then E is “not more complicated” than F, and the quotient space X/E hasa lesser or equal “Borel cardinality” than Y/F, where “Borel cardinality” is like cardinality except for a definabilityrestriction on the witnessing mapping.

26.2 Kuratowski’s theorem

A measure space X is called a standard Borel space if it is Borel-isomorphic to a Borel subset of a Polish space.Kuratowski’s theorem then states that two standard Borel spaces X and Y are Borel-isomorphic iff |X| = |Y |.

26.3 References• Harrington, L. A., A. S. Kechris, A. Louveau (Oct 1990). “A Glimm-Effros Dichotomy for Borel equivalencerelations”. Journal of the American Mathematical Society (Journal of the American Mathematical Society, Vol.3, No. 4) 3 (2): 903–928. doi:10.2307/1990906. JSTOR 1990906.

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.

• Silver, Jack H. (1980). “Counting the number of equivalence classes of Borel and coanalytic equivalencerelations”. Annals of Mathematical Logic 18 (1): 1–28. doi:10.1016/0003-4843(80)90002-9.

• Kanovei, Vladimir; Borel equivalence relations. Structure and classification. University Lecture Series, 44.American Mathematical Society, Providence, RI, 2008. x+240 pp. ISBN 978-0-8218-4453-3

30

Chapter 27

Cabal (set theory)

The Cabal was, or perhaps is, a grouping of set theorists in Southern California, particularly at UCLA and Caltech,but also at UC Irvine. Organization and procedures range from informal to nonexistent, so it is difficult to say whetherit still exists or exactly who has been a member, but it has included such notable figures as Donald A. Martin, YiannisN. Moschovakis, John R. Steel, and Alexander S. Kechris. Others who have published in the proceedings of theCabal seminar include Robert M. Solovay, W. Hugh Woodin, Matthew Foreman, and Steve Jackson.The work of the group is characterized by free use of large cardinal axioms, and research into the descriptive settheoretic behavior of sets of reals if such assumptions hold.Some of the philosophical views of the Cabal seminar were described in Maddy 1988a and Maddy 1988b.

27.1 Publications• Kechris, A. S.; et al. (1978). Cabal Seminar 76-77: Proceedings. Caltech-UCLA Logic Seminar 1976-77.Springer. ISBN 0-387-09086-X.

• Kechris, A. S. (editor) (1983). Cabal Seminar 79-81: Proc Caltech-UCLA Logic Seminar 1979-81 (LectureNotes in Mathematics). Springer. ISBN 0-387-12688-0.

• Martin, D. A., A. S. Kechris, J. R. Steel (1988). Cabal Seminar 81-85: Proceedings Caltech UCLA LogicSeminar (Lecture Notes in Mathematics, No 1333). Springer. ISBN 0-387-50020-0.

• Alexander S. Kechris, Benedikt Löwe, John R. Steel (2008). Games, Scales, and Suslin cardinals: The CabalSeminar Volume I: Lecture Notes in Logic. CUP. ISBN 9780521899512.

27.2 References• Maddy, Penelope (1988). “Believing the Axioms I” (PDF). The Journal of Symbolic Logic 53 (2): 481–511.doi:10.1017/s0022481200028425.

• Maddy, Penelope (1988). “Believing the Axioms II” (PDF). The Journal of Symbolic Logic 53 (3): 736–764.doi:10.2307/2274569.

31

Chapter 28

Cantor–Dedekind axiom

In mathematical logic, the phraseCantor–Dedekind axiom has been used to describe the thesis that the real numbersare order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to onecorrespondence between real numbers and points on a line.This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartesexplicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or planeinto a conceptual metaphor. This is sometimes referred to as the real number line blend:[1]

A consequence of this axiom is that Alfred Tarski’s proof of the decidability of the ordered real field could be seenas an algorithm to solve any problem in Euclidean geometry.

28.1 Notes[1] George Lakoff and Rafael E. Núñez (2000). Where Mathematics Comes From: How the embodied mind brings mathematics

into being. Basic Books. ISBN 0-465-03770-4.

28.2 References• Ehrlich, P. (1994). “General introduction”. Real Numbers, Generalizations of the Reals, and Theories of

Continua, vi–xxxii. Edited by P. Ehrlich, Kluwer Academic Publishers, Dordrecht

• Bruce E. Meserve (1953) Fundamental Concepts of Algebra, p. 32, at Google Books

• B.E. Meserve (1955) Fundamental Concepts of Geometry, p. 86, at Google Books

32

Chapter 29

Categorical set theory

Categorical set theory is any one of several versions of set theory developed from or treated in the context ofmathematical category theory.

29.1 References• Barr, M. and Wells, C., Category Theory for Computing Science, Hemel Hempstead, UK, 1990.

• Bourbaki, N., Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Ger-many, 1994.

• Kelley, J.L., General Topology, Van Nostrand Reinhold, New York, NY, 1955.

• Lambek, J. and Scott, P.J., Introduction to Higher Order Categorical Logic, Cambridge University Press, Cam-bridge, UK, 1986.

• Lawvere, F.W., and Rosebrugh, R., Sets for Mathematics, Cambridge University Press, Cambridge, UK, 2003.

• Lawvere, F.W., and Schanuel, S.H., Conceptual Mathematics, A First Introduction to Categories, CambridgeUniversity Press, Cambridge, UK, 1997. Reprinted with corrections, 2000.

• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.),MIT Press, Cambridge, MA, 1993.

• Mitchell, J.C., Foundations for Programming Languages, MIT Press, Cambridge, MA, 1996.

• Nestruev, J., Smooth Manifolds and Observables, Springer-Verlag, New York, NY, 2003. ISBN 0-387-95543-7.

• Poizat, B., A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Moses Klein(trans.), Springer-Verlag, New York, NY, 2000.

29.2 See also• Categorical logic

29.3 External links• Rethinking set theory by Tom Leinster

33

Chapter 30

Centered set

In mathematics, in the area of order theory, an upwards centered set S is a subset of a partially ordered set, P, suchthat any finite subset of S has an upper bound in P. Similarly, any finite subset of a downwards centered set has alower bound. An upwards centered set can also be called a consistent set. Note that any directed set is necessarilycentered, and any centered set is linked.A subset B of a partial order is said to be σ-centered if it is a countable union of centered sets.

30.1 References• Fremlin, David H. (1984). Consequences of Martin’s axiom. Cambridge tracts in mathematics, no. 84. Cam-bridge: Cambridge University Press. ISBN 0-521-25091-9.

• Davey, B. A.; Priestley, Hilary A. (2002), “9.1”, Introduction to Lattices and Order (2nd ed.), CambridgeUniversity Press, p. 201, ISBN 978-0-521-78451-1, Zbl 1002.06001.

34

Chapter 31

Chang’s conjecture

In model theory, a branch of mathematical logic, Chang’s conjecture, attributed to Chen Chung Chang by Vaught(1963, p. 309), states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type(ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinalityβ. The usual notation is (ω2, ω1) ↠ (ω1, ω) .The axiom of constructibility implies that Chang’s conjecture fails. Silver proved the consistency of Chang’s con-jecture from the consistency of an ω1-Erdős cardinal. Hans-Dieter Donder showed the reverse implication: if CCholds, then ω2 is ω1-Erdős in K.More generally, Chang’s conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β)for a countable language has an elementary submodel of type (γ,δ). The consistency of (ω3, ω2) ↠ (ω2, ω1) wasshown by Laver from the consistency of a huge cardinal.

31.1 References• Chang, Chen Chung; Keisler, H. Jerome (1990), Model Theory, Studies in Logic and the Foundations ofMathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3

• Vaught, R. L. (1963), “Models of complete theories”, Bulletin of the American Mathematical Society 69: 299–313, doi:10.1090/S0002-9904-1963-10903-9, ISSN 0002-9904, MR 0147396

35

Chapter 32

Church–Kleene ordinal

Inmathematics, theChurch–Kleene ordinal,ωCK1 , named after AlonzoChurch and S. C. Kleene, is a large countable

ordinal. It is the set of all recursive ordinals and the smallest non-recursive ordinal. It is also the first ordinal whichis not hyperarithmetical, and the first admissible ordinal after ω.

32.1 References• Church, Alonzo; Kleene, S. C. (1937), “Formal definitions in the theory of ordinal numbers.”, Fundamenta

mathematicae, Warszawa, 28: 11–21, JFM 63.0029.02

• Church, Alonzo (1938), “The constructive second number class”, Bull. Amer. Math. Soc. 44 (4): 224–232,doi:10.1090/S0002-9904-1938-06720-1

• Kleene, S. C. (1938), “On Notation for Ordinal Numbers”, The Journal of Symbolic Logic (The Journal ofSymbolic Logic, Vol. 3, No. 4) 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778

• Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT presspaperback edition, ISBN 978-0-262-68052-3

36

Chapter 33

Class logic

Class logic is a logic in its broad sense, whose objects are called classes. In a narrower sense, one speaks of a classof logic only if classes are described by a property of their elements. This class logic is thus a generalization of settheory, which allows only a limited consideration of classes.

33.1 Class logic in the strict sense

The first class logic in the strict sense was created by Giuseppe Peano in 1889 as the basis for his arithmetic (PeanoAxioms). He introduced the class term, which formally correctly describes classes through a property of their ele-ments. Today the class term is denoted in the form x|A(x), where A(x) is an arbitrary statement, which all classmembers x meet. Peano axiomatized the class term for the first time and used it fully. Gottlob Frege also triedestablishing the arithmetic logic with class terms in 1893; Bertrand Russell discovered a conflict in it in 1902 whichbecame known as Russell’s paradox. As a result, it became generally known that you can not safely use class terms.To solve the problem, Russell developed his type theory from 1903 to 1908, which allowed only a verymuch restricteduse of class terms. In the long term she not prevailed but, but more comfortable and more powerful, 1907 initiated byErnst Zermelo set theory. Not a class logic in the narrower sense, but in its present form (ZF or NBG) because it doesnot axiomatize the class term, but used only in practice as a useful notation. Willard Van Orman Quine described aset theory New Foundations (NF) in 1937, oriented not at Cantor, or Zermelo-Fraenkel, but on the theory of types.In 1940 Quine advanced NF to Mathematical Logic (ML). Since the antinomy of Burali-Forti was derived in the firstversion of ML,[1] Quine clarified ML, retaining the widespread use of classes, and took up a proposal by HaoWang[2]introducing in 1963 in his theory of x|A(x) as a virtual class, so that classes are although not yet full-fledged terms,but sub-terms in defined contexts.[3]

After Quine, Arnold Oberschelp developed the first fully functional modern axiomatic class logic starting in 1974. Itis a consistent extension of predicate logic and allows the unrestricted use of class terms (such as Peano).[4] It uses allclasses that produce antinomies of naive set theory as a term. This is possible because the theory assumes no existenceaxioms for classes. It presupposes in particular any number of axioms, but can also take those and syntactically correctto formulate in the traditionally simple design with class terms. For example, the Oberschelp set theory developedthe Zermelo–Fraenkel set theory within the framework of class logic.[5] Three principles guarantee that cumbersomeZF formulas are translatable into convenient classes formulas; guarantee a class logical increase in the ZF languagethey form without quantities axioms together with the axioms of predicate logic an axiom system for a simple logicof general class.[6]

The principle of abstraction (Abstraktionsprinzip) states that classes describe their elements via a logical property:

∀y : (y ∈ x | A(x) ⇐⇒ A(y))

The principle of extensionality (Extensionalitätsprinzip ) describes the equality of classes by matching their elementsand eliminates the axiom of extensionality in ZF:

37

38 CHAPTER 33. CLASS LOGIC

A = B ⇐⇒ ∀x : (x ∈ A ⇐⇒ x ∈ B)

The principle of comprehension (Komprehensionsprinzip) determines the existence of a class as an element:

x | A(x) ∈ B ⇐⇒ ∃y : (y = x | A(x) ∧ y ∈ B)

33.2 Bibliography• Giuseppe Peano: Arithmetices principia. Nova methodo exposita. Corso, Torino u. a. 1889 (Auch in: GiuseppePeano: Opere scelte. Band 2. Cremonese, Rom 1958, S. 20–55).

• G. Frege: Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet. Band 1. Pohle, Jena 1893.

• Willard Van Orman Quine: New Foundations for Mathematical Logic, in: American Mathematical Monthly44 (1937), S. 70-80.

• WillardVanOrmanQuine: Set Theory and its Logic. HarvardUniversity Press, CambridgeMA1963 (DeutscheÜbersetzung: Mengenlehre und ihre Logik (= Logik und Grundlagen der Mathematik. Bd. 10). Vieweg, Braun-schweig 1973, ISBN 3-548-03532-9).

• Arnold Oberschelp: Elementare Logik und Mengenlehre (= BI-Hochschultaschenbücher 407–408). 2 Bände.Bibliographisches Institut, Mannheim u. a. 1974–1978, ISBN 3-411-00407-X (Bd. 1), ISBN 3-411-00408-8(Bd. 2).

• Albert Menne Grundriß der formalen Logik (= Uni-Taschenbücher 59 UTB für Wissenschaft). Schöningh,Paderborn 1983, ISBN 3-506-99153-1 (Renamed Grundriß der Logistik starting with 5th Edition – The bookshows, among other calcului, a possible application of calculus to class logic, based on the propositional andpredicate calculus and carried the basic terms of formal systems to class logic. It also discusses briefly theparadoxes and type theory).

• Jürgen-Michael Glubrecht, ArnoldOberschelp, Günter Todt: Klassenlogik. Bibliographisches Institut, Mannheimu. a. 1983, ISBN 3-411-01634-5.

• Arnold Oberschelp: Allgemeine Mengenlehre. BI-Wissenschafts-Verlag, Mannheim u. a. 1994, ISBN 3-411-17271-1.

33.3 References[1] John Barkley Rosser: Burali-Forti paradox. In: Journal of Symbolic Logic, Band 7, 1942, p. 1-17

[2] Hao Wang: A formal system for logic. In: Journal of Symbolic Logic, Band 15, 1950, p. 25-32

[3] Willard Van Orman Quine: Mengenlehre und ihre Logik. 1973, S. 12.

[4] Arnold Oberschelp: Allgemeine Mengenlehre. 1994, p. 75 f.

[5] The advantages of the class logic are shown in a comparison of ZFC in class logic and predicate logic form in: ArnoldOberschelp: Allgemeine Mengenlehre. 1994, p. 261.

[6] Arnold Oberschelp, p. 262, 41.7. The axiomatization is much more complicated, but here is reduced to a book-end to theessentials.

Chapter 34

Classical mathematics

In the foundations ofmathematics, classicalmathematics refers generally to themainstream approach tomathematics,which is based on classical logic and ZFC set theory.[1] It stands in contrast to other types of mathematics such asconstructive mathematics or predicative mathematics. In practice, the most common non-classical systems are usedin constructive mathematics.[2]

Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist and other objections tothe logic, set theory, etc., chosen as its foundations, such as have been expressed by L. E. J. Brouwer. Almost allmathematics, however, is done in the classical tradition, or in ways compatible with it.Defenders of classical mathematics, such as David Hilbert, have argued that it is easier to work in, and is most fruitful;although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematicscould not (or could not so easily) attain, they argue that on the whole, it is the other way round.In terms of the philosophy and history of mathematics, the very existence of non-classical mathematics raises thequestion of the extent to which the foundational mathematical choices humanity has made arise from their “superi-ority” rather than from, say, expedience-driven concentrations of effort on particular aspects.

34.1 See also• Constructivism (mathematics)

• Finitism

• Intuitionism

• Non-classical analysis

• Traditional mathematics

• Ultrafinitism

• Philosophy of Mathematics

34.2 References[1] Stewart Shapiro, ed. (2005). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press,

USA. ISBN 978-0-19-514877-0.

[2] Torkel Franzén (1987). Provability and Truth. Almqvist & Wiksell International. ISBN 91-22-01158-7.

39

Chapter 35

Coanalytic set

In the mathematical discipline of descriptive set theory, a coanalytic set is a set (typically a set of real numbers ormore generally a subset of a Polish space) that is the complement of an analytic set (Kechris 1994:87). Coanalyticsets are also referred to as Π1

1 sets (see projective hierarchy).

35.1 References• Kechris, Alexander S. (1994), Classical Descriptive Set Theory, Springer-Verlag, ISBN 0-387-94374-9

40

Chapter 36

Cocountability

In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In otherwords, Y contains all but countably many elements of X. While the rational numbers are a countable subset of thereals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then onesays Y is cofinite.

36.1 σ-algebras

The set of all subsets of X that are either countable or cocountable forms a σ-algebra, i.e., it is closed under theoperations of countable unions, countable intersections, and complementation. This σ-algebra is the countable-cocountable algebra on X. It is the smallest σ-algebra containing every singleton set.

36.2 Topology

The cocountable topology (also called the “countable complement topology”) on any set X consists of the empty setand all cocountable subsets of X.

41

Chapter 37

Code (set theory)

In set theory, a code for a hereditarily countable set

x ∈ Hℵ1

is a set

E ⊂ ω × ω

such that there is an isomorphism between (ω,E) and (X, ∈ ) where X is the transitive closure of x. If X is finite(with cardinality n), then use n×n instead of ω×ω and (n,E) instead of (ω,E).According to the axiom of extensionality, the identity of a set is determined by its elements. And since those elementsare also sets, their identities are determined by their elements, etc.. So if one knows the element relation restrictedto X, then one knows what x is. (We use the transitive closure of x rather than of x itself to avoid confusing theelements of x with elements of its elements or whatever.) A code includes that information identifying x and alsoinformation about the particular injection from X into ω which was used to create E. The extra information about theinjection is non-essential, so there are many codes for the same set which are equally useful.So codes are a way of mapping Hℵ1 into the powerset of ω×ω. Using a pairing function on ω (such as (n,k) goes to(n2+2·n·k+k2+n+3·k)/2), we can map the powerset of ω×ω into the powerset of ω. And we can map the powerset ofω into the Cantor set, a subset of the real numbers. So statements aboutHℵ1 can be converted into statements aboutthe reals. Consequently,Hℵ1 ⊂ L(R) .

Codes are useful in constructing mice.

37.1 See also• L(R)

37.2 References• William J.Mitchell,"The Complexity of the CoreModel”,"Journal of Symbolic Logic”,Vol.63,No.4,December1998,page 1393.

42

Chapter 38

Coherent space

In proof theory, a coherent space is a concept introduced in the semantic study of linear logic.Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton. Fora family of C-sets (i.e., F ⊆ ℘(C)), the dual of F, written F ⊥, is defined as the set of all C-sets S such that for everyT ∈ F, S ⊥ T. A coherent space F over C is a family C-sets for which F = (F ⊥) ⊥.In topology, a coherent space is another name for spectral space. A continuous map between coherent spaces iscalled coherent if it is spectral.In Proofs and Types coherent spaces are called coherence spaces. A footnote explains that although in the Frenchoriginal they were espaces cohérents, the coherence space translation was used because spectral spaces are sometimescalled coherent spaces.

38.1 References• Girard, J.-Y.; Lafont, Y.; Taylor, P. (1989), Proofs and types, Cambridge University Press.

• Girard, J.-Y. (2004), “Between logic and quantic: a tract”, in Ehrhard; Girard; Ruet; et al., Linear logic incomputer science (PDF), Cambridge University Press.

• Johnstone, Peter (1982), “II.3 Coherent locales”, Stone Spaces, Cambridge University Press, pp. 62–69, ISBN978-0-521-33779-3.

43

Chapter 39

Complete theory

In mathematical logic, a theory is complete if it is amaximal consistent set of sentences, i.e., if it is consistent, andnone of its proper extensions is consistent. For theories in logics which contain classical propositional logic, this isequivalent to asking that for every sentence φ in the language of the theory it contains either φ itself or its negation¬φ.Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to beformulated cannot be complete, as demonstrated by Gödel’s incompleteness theorem.This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can beformulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of “seman-tically valid”). Gödel’s completeness theorem is about this latter kind of completeness.Complete theories are closed under a number of conditions internally modelling the T-schema:

• For a set S : A ∧B ∈ S if and only if A ∈ S and B ∈ S ,

• For a set S : A ∨B ∈ S if and only if A ∈ S or B ∈ S .

Maximal consistent sets are a fundamental tool in the model theory of classical logic and modal logic. Their existencein a given case is usually a straightforward consequence of Zorn’s lemma, based on the idea that a contradictioninvolves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent setsextending a theory T (closed under the necessitation rule) can be given the structure of a model of T, called thecanonical model.

39.1 Examples

Some examples of complete theories are:

• Presburger arithmetic

• Tarski’s axioms for Euclidean geometry

• The theory of dense linear orders

• The theory of algebraically closed fields of a given characteristic

• The theory of real closed fields

• Every uncountably categorical countable theory

• Every countably categorical countable theory

44

39.2. REFERENCES 45

39.2 References• Mendelson, Elliott (1997). Introduction to Mathematical Logic (Fourth ed.). Chapman & Hall. p. 86. ISBN978-0-412-80830-2.

Chapter 40

Completeness of atomic initial sequents

In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A isan arbitrary formula) can be derived from only atomic initial sequents p ⊢ p (where p is an atomic formula). Thistheorem plays a role analogous to eta expansion in lambda calculus, and dual to cut-elimination and beta reduction.Typically it can be established by induction on the structure of A, much more easily than cut-elimination.

40.1 References• Gaisi Takeuti. Proof theory. Volume 81 of Studies in Logic and the Foundation ofMathematics. North-Holland,Amsterdam, 1975.

• Anne Sjerp Troelstra and Helmut Schwichtenberg. Basic Proof Theory. Edition: 2, illustrated, revised. Pub-lished by Cambridge University Press, 2000.

46

Chapter 41

Computable isomorphism

In computability theory two sets A;B ⊆ N of natural numbers are computably isomorphic or recursively isomor-phic if there exists a total bijective computable function f : N → N with f(A) = B . By the theorem of Myhill,[1]the relation of computable isomorphism coincides with the relation of one-one reduction.Two numberings ν and µ are called computably isomorphic if there exists a computable bijection f so that ν = µfComputably isomorphic numberings induce the same notion of computability on a set.

41.1 References[1] Theorem 7.VI, Hartley Rogers, Jr., Theory of recursive functions and effective computability

• Rogers, Hartley, Jr. (1987), Theory of recursive functions and effective computability (2nd ed.), Cambridge,MA: MIT Press, ISBN 0-262-68052-1, MR 886890.

47

Chapter 42

Computable measure theory

In mathematics, computable measure theory is the part of computable analysis that deals with effective versions ofmeasure theory.

42.1 References• Jeremy Avigad (2012), “Inverting the Furstenberg correspondence”, Discrete and Continuous Dynamical Sys-

tems, Series A, 32, pp. 3421–3431.

• Abbas Edalat (2009), “A computable approach to measure and integration theory”, Information and Compu-tation 207:5, pp. 642–659.

• Stephen G. Simpson (2009), Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, CambridgeUniversity Press. ISBN 978-0-521-88439-6

48

Chapter 43

Computable model theory

Computable model theory is a branch of model theory which deals with questions of computability as they applyto model-theoretical structures.

43.1 History

It was developed almost simultaneously by mathematicians in the West, primarily located in the United States andAustralia, and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little com-munication between these two groups and so a number of important results were discovered independently.

43.2 Introduction

Computable model theory introduces the ideas of computable and decidable models and theories and one of the basicproblems is discovering whether or not computable or decidable models fulfilling certain model-theoretic conditionscan be shown to exist.

43.3 See also• Vaught conjecture

43.4 References• Harizanov, V. S. (1998), “Pure Computable Model Theory”, in Ershov, Iurii Leonidovich, Handbook of Re-

cursive Mathematics, Volume 1: Recursive Model Theory, Studies in Logic and the Foundations of Mathematics138, North Holland, pp. 3–114, ISBN 978-0-444-50003-8, MR 1673621.

49

Chapter 44

Computable real function

In mathematical logic, specifically computability theory, a function f : R → R is sequentially computable if, for everycomputable sequence xi∞i=1 of real numbers, the sequence f(xi)∞i=1 is also computable.A function f : R → R is effectively uniformly continuous if there exists a recursive function d : N → N such that, if|x− y| < 1

d(n)

then|f(x)− f(y)| < 1

n

A real function is computable if it is both sequentially computable and effectively uniformly continuous,[1]

These definitions can be generalized to functions of more than one variable or functions only defined on a subset ofRn. The generalizations of the latter two need not be restated. A suitable generalization of the first definition is:LetD be a subset ofRn.A function f : D → R is sequentially computable if, for everyn -tuplet (xi 1∞i=1, . . . xi n∞i=1)of computable sequences of real numbers such that(∀i) (xi 1, . . . xi n) ∈ D ,

the sequence f(xi)∞i=1 is also computable.This article incorporates material from Computable real function on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

44.1 References[1] see Grzegorczyk, Andrzej (1957), “On the Definitions of Computable Real Continuous Functions” (PDF), Fundamenta

Mathematicae 44: 61–77

50

Chapter 45

Condensation lemma

In set theory, a branch of mathematics, the condensation lemma is a result about sets in the constructible universe.It states that if X is a transitive set and is an elementary submodel of some level of the constructible hierarchy Lα,that is, (X,∈) ≺ (Lα,∈) , then in fact there is some ordinal β ≤ α such that X = Lβ .More can be said: If X is not transitive, then its transitive collapse is equal to some Lβ , and the hypothesis ofelementarity can be weakened to elementarity only for formulas which are Σ1 in the Lévy hierarchy. Also, theassumption that X be transitive automatically holds when α = ω1 .The lemma was formulated and proved by Kurt Gödel in his proof that the axiom of constructibility implies GCH.

45.1 References• Devlin, Keith (1984). Constructibility. Springer. ISBN 3-540-13258-9. (theorem II.5.2 and lemma II.5.10)

51

Chapter 46

Conglomerate (set theory)

In mathematics, a conglomerate is a definable collection of classes, just as a class is a definable collection of sets.[1]A quasi-category is like a category except that its objects and morphisms form conglomerates instead of classes.[1]The subclasses of any class, and in particular, the collection of all classes (every class is a subclass of the class of allsets), form a conglomerate.

46.1 References[1] Adamek, Jiri; Herrlich, Horst; Strecker, George (1990). Abstract and Concrete Categories: The Joy of Cats. Dover Publi-

cations. ISBN 978-0-486-46934-8.

52

Chapter 47

Conservativity theorem

In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula

∃x1 . . .∃xm φ(x1, . . . , xm)

is a theorem of a first-order theory T . Let T1 be a theory obtained from T by extending its language with newconstants

a1, . . . , am

and adding a new axiom

φ(a1, . . . , am)

Then T1 is a conservative extension of T , which means that the theory T1 has the same set of theorems in the originallanguage (i.e., without constants ai ) as the theory T .In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by intro-ducing a new functional symbol:

Suppose that a closed formula ∀y ∃xφ(x, y) is a theorem of a first-order theory T , where we denotey := (y1, . . . , yn) . Let T1 be a theory obtained from T by extending its language with new functionalsymbol f (of arity n ) and adding a new axiom ∀y φ(f(y), y) . Then T1 is a conservative extension ofT , i.e. the theories T and T1 prove the same theorems not involving the functional symbol f ).

47.1 References• Elliott Mendelson (1997). Introduction to Mathematical Logic (4th ed.) Chapman & Hall.

• J.R. Shoenfield (1967). Mathematical Logic. Addison-Wesley Publishing Company.

53

Chapter 48

Constructive non-standard analysis

In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's non-standard analysis,developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote:

The possibility of constructivization of nonstandard analysis was studied by Palmgren (1997, 1998,2001). Themodel of constructive nonstandard analysis studied there is an extension ofMoerdijk’s (1995)model for constructive nonstandard arithmetic.

48.1 See also• Smooth infinitesimal analysis

• John Lane Bell

48.2 References• Ieke Moerdijk, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73(1995), pp. 37–51.

“Abstract: This paper provides an explicit description of a model for intuitionistic non-standard arith-metic, which can be formalized in a constructive metatheory without the axiom of choice.”

• Erik Palmgren, Developments in Constructive Nonstandard Analysis, Bull. Symbolic Logic Volume 4, Number3 (1998), 233–272.

“Abstract: We develop a constructive version of nonstandard analysis, extending Bishop's constructiveanalysis with infinitesimal methods. ...”

• Juha Ruokolainen 2004, Constructive Nonstandard Analysis Without Actual Infinity

54

Chapter 49

Continuous function (set theory)

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumedat limit stages are the limits (limit suprema and limit infima) of all values at previous stages. More formally, let γ bean ordinal, and s := ⟨sα|α < γ⟩ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ,

sβ = lim supsα|α < β = infsupsα|δ ≤ α < β|δ < β

and

sβ = lim infsα|α < β = supinfsα|δ ≤ α < β|δ < β .

Alternatively, s is continuous if s: γ → range(s) is a continuous function when the sets are each equipped with theorder topology. These continuous functions are often used in cofinalities and cardinal numbers.

49.1 References• Thomas Jech. Set Theory, 3rd millennium ed., 2002, Springer Monographs in Mathematics,Springer, ISBN3-540-44085-2

55

Chapter 50

Continuum (set theory)

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite)cardinal number, c . Georg Cantor proved that the cardinality c is larger than the smallest infinity, namely, ℵ0 . Healso proved that c equals 2ℵ0 , the cardinality of the power set of the natural numbers.The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes statedby saying that no cardinality lies between that of the continuum and that of the natural numbers, ℵ0 .

50.1 Linear continuum

Main article: Linear continuum

According to Raymond Wilder (1965) there are four axioms that make a set C and the relation < into a linearcontinuum:

• C is simply ordered with respect to <.

• If [A,B] is a cut of C, then either A has a last element or B has a first element. (compare Dedekind cut)

• There exists a non-empty, countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ Ssuch that x < z < y. (separability axiom)

• C has no first element and no last element. (Unboundedness axiom)

These axioms characterize the order type of the real number line.

50.2 See also• Suslin’s problem

50.3 References• Raymond L. Wilder (1965) The Foundations of Mathematics, 2nd ed., page 150, John Wiley & Sons.

56

Chapter 51

Continuum function

In mathematics, the continuum function is κ 7→ 2κ , i.e. raising 2 to the power of κ using cardinal exponentiation.Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality.

51.1 See also• Continuum hypothesis

• Cardinality of the continuum

• Beth number

• Easton’s theorem

• Gimel function

57

Chapter 52

Conull set

In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero.[1] Forexample, the set of irrational numbers is a conull subset of the real line with Lebesgue measure.[2]

A property that is true of the elements of a conull set is said to be true almost everywhere.[3]

52.1 References[1] Führ, Hartmut (2005), Abstract harmonic analysis of continuous wavelet transforms, Lecture Notes in Mathematics 1863,

Springer-Verlag, Berlin, p. 12, ISBN 3-540-24259-7, MR 2130226.

[2] A related but slightly more complex example is given by Führ, p. 143.

[3] Bezuglyi, Sergey (2000), “Groups of automorphisms of a measure space and weak equivalence of cocycles”, Descriptiveset theory and dynamical systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser. 277, Cambridge Univ.Press, Cambridge, pp. 59–86, MR 1774424. See p. 62 for an example of this usage.

58

Chapter 53

Countryman line

ACountryman line is an uncountable linear ordering whose square is the union of countably many chains. The exis-tence of Countryman lines was first proven by Shelah. Shelah also conjectured that, assuming PFA, every Aronszajnline contains a Countryman line. This conjecture, which remained open for three decades, was proven by JustinMoore.

53.1 References• Shelah, Saharon (1976). “Decomposing uncountable squares to countably many chains”. Journal of Combi-

natorial Theory Series A 21 (1): 110–114. doi:10.1016/0097-3165(76)90053-4.

• Moore, Justin (2006). “A five element basis for the uncountable linear orders”. Annals of Mathematics. SecondSeries 163 (2): 669–688. doi:10.4007/annals.2006.163.669.

59

Chapter 54

Critical point (set theory)

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is thesmallest ordinal which is not mapped to itself.[1]

Suppose that j : N →M is an elementary embedding where N andM are transitive classes and j is definable in N by aformula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing.Also j(ω)=ω. If j(α)=α for all α<κ and j(κ)>κ, then κ is said to be the critical point of j.If N is V, then κ (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number κsuch that there exists a <κ-complete, non-principal ultrafilter over κ. Specifically, one may take the filter to beA|A ⊆ κ∧κ ∈ j(A) .Generally, there will bemany other <κ-complete, non-principal ultrafilters over κ. However,j might be different from the ultrapower(s) arising from such filter(s).If N and M are the same and j is the identity function on N, then j is called “trivial”. If transitive class N is an innermodel of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial.

54.1 References[1] Jech, Thomas (2002). Set Theory. Berlin: Springer-Verlag. ISBN 3-540-44085-2. p. 323

60

Chapter 55

Cumulative hierarchy

In mathematical set theory, a cumulative hierarchy is a family of setsWα indexed by ordinals α such that

• Wα⊆Wα₊₁

• If α is a limit thenWα = ∪ᵦ<αWᵦ

It is also sometimes assumed thatWα₊₁⊆P(Wα) or thatW0 is empty.The unionW of the sets of a cumulative hierarchy is often used as a model of set theory.The phrase “the cumulative hierarchy” usually refers to the standard cumulative hierarchy Vα of the Von Neumannuniverse with Vα₊₁=P(Vα).

55.1 Reflection principle

A cumulative hierarchy satisfies a form of the reflection principle: any formula of the language of set theory thatholds in the unionW of the hierarchy also holds in some stagesWα.

55.2 Examples• The Von Neumann universe is built from a cumulative hierarchy Vα.

• The sets Lα of the constructible universe form a cumulative hierarchy.

• The Boolean valued models constructed by forcing are built using a cumulative hierarchy.

• The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation) form a cumu-lative hierarchy whose union satisfies the axiom of foundation.

55.3 References• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

61

Chapter 56

Cyclic negation

In many-valued logic with linearly ordered truth values, cyclic negation is a unary truth function that takes a truthvalue n and returns n − 1 as value if n isn't the lowest value; otherwise it returns the highest value.For example, let the set of truth values be 0,1,2, let ~ denote negation, and let p be a variable ranging over truthvalues. For these choices, if p = 0 then ~p = 2; and if p = 1 then ~p = 0.Cyclic negation was originally introduced by the logician and mathematician Emil Post.

56.1 References• Mares, Edwin (2011), “Negation”, in Horsten, Leon; Pettigrew, Richard, The Continuum Companion to Philo-

sophical Logic, Continuum International Publishing, pp. 180–215, ISBN 9781441154231. See in particularpp. 188–189.

62

Chapter 57

Dense order

In mathematics, a partial order or total order < on a set X is said to be dense if, for all x and y in X for which x < y,there is a z in X such that x < z < y.

57.1 Example

The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. Onthe other hand, the ordinary ordering on the integers is not dense.

57.2 Uniqueness

Georg Cantor proved that every two densely totally ordered countable sets without lower or upper bounds are order-isomorphic.[1] In particular, there exists an isomorphism between the rational numbers and other densely orderedcountable sets including the dyadic rationals and the algebraic numbers. The proof of this result uses the back-and-forth method.[2]

Minkowski’s questionmark function can be used to determine the order isomorphisms between the quadratic algebraicnumbers and the rational numbers, and between the rationals and the dyadic rationals.

57.3 Generalizations

Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y areR-related. Formally:

∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).

Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.

57.4 See also

• Dense set

• Dense-in-itself

• Kripke semantics

63

64 CHAPTER 57. DENSE ORDER

57.5 References[1] Roitman, Judith (1990), “Theorem 27, p. 123”, Introduction to Modern Set Theory, Pure and Applied Mathematics 8, John

Wiley & Sons, ISBN 9780471635192.

[2] Dasgupta, Abhijit (2013), Set Theory: With an Introduction to Real Point Sets, Springer-Verlag, p. 161, ISBN9781461488545.

57.6 Additional reading• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff

Chapter 58

Deviation of a poset

In order-theoretic mathematics, the deviation of a poset is an ordinal number measuring the complexity of a partiallyordered set.The deviation of a poset is used to define the Krull dimension of a module over a ring as the deviation of its poset ofsubmodules.

58.1 Definition

A poset is said to have deviation at most α (for an ordinal α) if for every descending chain of elements a0 > a1 >...all but a finite number of the posets of elements between an and an₊₁ have deviation less than α. The deviation (if itexists) is the minimum value of α for which this is true.Not every poset has a deviation. The following conditions on a poset are equivalent:

• The poset has a deviation

• The opposite poset has a deviation

• The poset does not contain a subset order-isomorphic to the rational numbers (with their standard numericalordering)

58.2 Example

The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviationis vacuously true. However, its opposite poset has deviation 1.

58.3 References• McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian rings, Graduate Studies in Mathematics30 (Revised ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2169-5, MR 1811901

65

Chapter 59

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If δ is an ordinal number and ⟨Xα | α < δ⟩ is a sequence of subsets of δ , then the diagonal intersection, denoted by

∆α<δXα,

is defined to be

β < δ | β ∈∩α<β

Xα.

That is, an ordinal β is in the diagonal intersection ∆α<δXα if and only if it is contained in the first β members ofthe sequence. This is the same as

∩α<δ

([0, α] ∪Xα),

where the closed interval from 0 to α is used to avoid restricting the range of the intersection.

59.1 See also• Fodor’s lemma

• Club set

• Club filter

59.2 References• Thomas Jech, Set Theory, The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003,page 92.

• Akihiro Kanamori, The Higher Infinite, Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.

This article incorporates material from diagonal intersection on PlanetMath, which is licensed under the Creative Com-mons Attribution/Share-Alike License.

66

Chapter 60

Difference hierarchy

In set theory, the difference hierarchy over a pointclass is a hierarchy of larger pointclasses generated by takingdifferences of sets. If Γ is a pointclass, then the set of differences in Γ is A : ∃C,D ∈ Γ(A = C \D) . In usualnotation, this set is denoted by 2-Γ. The next level of the hierarchy is denoted by 3-Γ and consists of differences ofthree sets: A : ∃C,D,E ∈ Γ(A = C \ (D \E)) . This definition can be extended recursively into the transfiniteto α-Γ for some ordinal α.[1]

In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the differencehierarchy over Π0ᵧ give Δ0ᵧ₊₁.[2]

60.1 References[1] Kanamori, Akihiro (2009), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, SpringerMonographs

in Mathematics (2nd ed.), Springer-Verlag, Berlin, p. 442, ISBN 978-3-540-88866-6, MR 2731169.

[2] Wadge, WilliamW. (2012), “Early investigations of the degrees of Borel sets”,Wadge degrees and projective ordinals. TheCabal Seminar. Volume II, Lect. Notes Log. 37, Assoc. Symbol. Logic, La Jolla, CA, pp. 166–195, MR 2906999. Seein particular p. 173.

67

Chapter 61

Double recursion

In recursive function theory, double recursion is an extension of primitive recursion which allows the definition ofnon-primitive recursive functions like the Ackermann function.Raphael M. Robinson called functions of two natural number variables G(n, x) double recursive with respect to givenfunctions, if

• G(0, x) is a given function of x.

• G(n + 1, 0) is obtained by substitution from the function G(n, ·) and given functions.

• G(n + 1, x + 1) is obtained by substitution from G(n + 1, x), the function G(n, ·) and given functions.[1]

Robinson goes on to provide a specific double recursive function (originally defined by Rózsa Péter)

• G(0, x) = x + 1

• G(n + 1, 0) = G(n, 1)

• G(n + 1, x + 1) = G(n, G(n + 1, x))

where the given functions are primitive recursive, butG is not primitive recursive. In fact, this is precisely the functionnow known as the Ackermann function.

61.1 See also• Primitive recursion

• Ackermann function

61.2 References[1] Raphael M. Robinson (1948). “Recursion and Double Recursion”. Bulletin of the American Mathematical Society 54:

987–93. doi:10.1090/S0002-9904-1948-09121-2.

68

Chapter 62

Double turnstile

Not to be confused with .

In logic, the symbol ⊨, ⊨ or |= is called the double turnstile. It is closely related to the turnstile symbol ⊢ , which hasa single bar across the middle. It is often read as "entails", "models", “is a semantic consequence of” or “is strongerthan”.[1] In TeX, the turnstile symbols ⊨ and |= are obtained from the commands \vDash and \models respectively.In Unicode it is encoded at U+22A8 ⊨ true (HTML &#8872;)In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and iscapable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial onusing this package.

62.1 Meaning

The double turnstile is a binary relation. It has several different meanings in different contexts:

• To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denotethat if every sentence on the left is true, the sentence on the right must be true, e.g. Γ ⊨ φ . This usage isclosely related to the single-barred turnstile symbol which denotes syntactic consequence.

• To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denotethat the structure is a model for (or satisfies) the set of sentences, e.g. A |= Γ .

• To denote a tautology, ⊨ φ . which is to say that the expression φ is a semantic consequence of the empty set.

62.2 See also• List of logic symbols

• List of mathematical symbols

62.3 References[1] Nederpelt, Rob (2004). “Chapter 7: Strengthening and weakening”. Logical Reasoning: A First Course (3rd revised ed.).

King’s College Publications. p. 62. ISBN 0-9543006-7-X.

69

Chapter 63

Effective descriptive set theory

Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightfacedefinitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effectivedescriptive set theory combines descriptive set theory with recursion theory.

63.1 Constructions

63.1.1 Effective Polish space

Main article: Effective Polish space

An effective Polish space is a complete separable metric space that has a computable presentation. Such spaces arestudied in both effective descriptive set theory and in constructive analysis. In particular, standard examples of Polishspaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

63.1.2 Arithmetical hierarchy

Main article: Arithmetical hierarchy

The arithmetical hierarchy, arithmetic hierarchy or Kleene-Mostowski hierarchy classifies certain sets based onthe complexity of formulas that define them. Any set that receives a classification is called arithmetical.More formally, the arithmetical hierarchy assigns classifications to the formulas in the language of first-order arith-metic. The classifications are denoted Σ0

n and Π0n for natural numbers n (including 0). The Greek letters here are

lightface symbols, which indicates that the formulas do not contain set parameters.If a formula ϕ is logically equivalent to a formula with only bounded quantifiers then ϕ is assigned the classificationsΣ0

0 and Π00 .

The classifications Σ0n and Π0

n are defined inductively for every natural number n using the following rules:

• If ϕ is logically equivalent to a formula of the form ∃n1∃n2 · · · ∃nkψ , where ψ is Π0n , then ϕ is assigned the

classification Σ0n+1 .

• If ϕ is logically equivalent to a formula of the form ∀n1∀n2 · · · ∀nkψ , where ψ is Σ0n , then ϕ is assigned the

classification Π0n+1 .

63.2 References• Mansfield, Richard; Weitkamp, Galen (1985). Recursive Aspects of Descriptive Set Theory. Oxford UniversityPress. pp. 124–38. ISBN 978-0-19-503602-2. MR 786122.

70

63.2. REFERENCES 71

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Secondedition available online

Chapter 64

Effective Polish space

In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presen-tation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standardexamples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.

64.1 Definition

An effective Polish space is a complete separable metric space X with metric d such that there is a countable denseset C = (c0, c1,...) that makes the following two relations on N4 computable (Moschovakis 2009:96-7):

P (i, j, k,m) ≡ d(ci, cj) ≤m

k + 1

Q(i, j, k,m) ≡ d(ci, cj) <m

k + 1

64.2 References• Yiannis N. Moschovakis, 2009, Descriptive Set Theory, 2nd edition, American Mathematical Society. ISBN0-8218-4813-5

72

Chapter 65

Elementary definition

In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic,and in particular without reference to set theory or using extensions such as plural quantification.Elementary definitions are of particular interest because they admit a complete proof apparatus while still beingexpressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms suchas ZFC).Saying that a definition is elementary is a weaker condition than saying it is algebraic.

65.1 Related• Elementary sentence

• Elementary theory

65.2 References• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

73

Chapter 66

Elementary diagram

For diagrams of electrical circuits, see Circuit diagram.

In the mathematical field of model theory, the elementary diagram of a structure is the set of all sentences withparameters from the structure that are true in the structure. It is also called the complete diagram.

66.1 Definition

Let M be a structure in a first-order language L. An extended language L(M) is obtained by adding to L a constantsymbol ca for every element a of M. The structure M can be viewed as an L(M) structure in which the symbols in Lare interpreted as before, and each new constant ca is interpreted as the element a. The elementary diagram of M isthe set of all L(M) sentences that are true in M (Marker 2002:44).

66.2 References• Chang, Chen Chung; Keisler, H. Jerome (1989), Model Theory, Elsevier, ISBN 978-0-7204-0692-4

• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6

• Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York:Springer-Verlag, ISBN 978-0-387-98760-6

74

Chapter 67

Elementary sentence

In mathematical logic, an elementary sentence is one that is stated using only finitary first-order logic, withoutreference to set theory or using any axioms which have consistency strength equal to set theory.Saying that a sentence is elementary is a weaker condition than saying it is algebraic.

67.1 Related• Elementary theory

• Elementary definition

67.2 References• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

75

Chapter 68

Elementary theory

In mathematical logic, an elementary theory is one that involves axioms using only finitary first-order logic, withoutreference to set theory or using any axioms which have consistency strength equal to set theory.Saying that a theory is elementary is a weaker condition than saying it is algebraic.

68.1 Related• Elementary sentence

• Elementary definition

• Elementary theory of the reals

68.2 References• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

76

Chapter 69

End extension

In model theory and set theory, which are disciplines within mathematics, a model B = ⟨B,F ⟩ of some axiomsystem of set theory T in the language of set theory is an end extension of A = ⟨A,E⟩ , in symbols A ⊆end B , if

• A is a substructure ofB , and

• b ∈ A whenever a ∈ A and bFa hold, i.e., no new elements are added byB to the elements of A .

The following is an equivalent definition of end extension: A is a substructure of B , and b ∈ A : bEa = b ∈B : bFa for all a ∈ A .For example, ⟨B,∈⟩ is an end extension of ⟨A,∈⟩ if A and B are transitive sets, and A ⊆ B .

77

Chapter 70

Epsilon-induction

In mathematics, ∈ -induction (epsilon-induction) is a variant of transfinite induction that can be used in set theoryto prove that all sets satisfy a given property P[x]. If the truth of the property for x follows from its truth for allelements of x, for every set x, then the property is true of all sets. In symbols:

∀x(∀y(y ∈ x→ P [y]) → P [x]

)→ ∀xP [x]

This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity giventhe other ZF axioms. ∈ -induction is a special case of well-founded induction.The name is most often pronounced “epsilon-induction”, because the set membership symbol∈ historically developedfrom the Greek letter ϵ .

70.1 See also• Mathematical induction

• Transfinite induction

• Well-founded induction

78

Chapter 71

Equisatisfiability

In logic, two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa;in other words, either both formulae are satisfiable or both are not. Two equisatisfiable formulae may have differentmodels, provided they both have some or both have none. As a result, equisatisfiability is different from logicalequivalence, as two equivalent formulae always have the same models.Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to becorrect if the original and resulting formulae are equisatisfiable. Examples of translations involving this concept areSkolemization and some translations into conjunctive normal form.

71.1 Examples

A translation from propositional logic into propositional logic in which every binary disjunction a ∨ b is replaced by((a ∨ n) ∧ (¬n ∨ b)) , where n is a new variable (one for each replaced disjunction) is a transformation in whichsatisfiability is preserved: the original and resulting formulae are equisatisfiable. Note that these two formulae are notequivalent: the first formula has the model in which b is true while a and n are false, and this is not a model of thesecond formula, in which n has to be true in this case.

79

Chapter 72

Erasure (logic)

In mathematical logic, a logical system has the erasure property if and only if no subset of the propositions can beadded to another subset of the propositions to refute a consequence.For instance, if proposition A means “the store is open from 8:00 to 22:00” and proposition B means “except Tues-days”, the system AB does not have erasure.

72.1 See also• Monotonic logic in “mathematical logic”

• Peirce’s Logic at the “Stanford Encyclopedia of Philosophy”

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Chapter 73

Erdős cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal numberintroduced by Paul Erdős and András Hajnal (1958).The Erdős cardinal κ(α) is defined to be the least cardinal such that for every function f : κ< ω → 0, 1, there is aset of order type α that is homogeneous for f (if such a cardinal exists). In the notation of the partition calculus, theErdős cardinal κ(α) is the smallest cardinal such that

κ(α) → (α)< ω

Existence of zero sharp implies that the constructible universe L satisfies “for every countable ordinal α, there is anα-Erdős cardinal”. In fact, for every indiscernible κ, Lκ satisfies “for every ordinal α, there is an α-Erdős cardinal inColl(ω, α) (the Levy collapse to make α countable)".However, existence of an ω1-Erdős cardinal implies existence of zero sharp. If f is the satisfaction relation for L(using ordinal parameters), then existence of zero sharp is equivalent to there being an ω1-Erdős ordinal with respectto f . And this in turn, the zero sharp implies the falsity of axiom of constructibility, of Kurt Gödel.If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable”.

73.1 References• Baumgartner, James E.; Galvin, Fred (1978). “Generalized Erdős cardinals and 0#". Annals of Mathematical

Logic 15 (3): 289–313. doi:10.1016/0003-4843(78)90012-8. ISSN 0003-4843. MR 528659

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations ofMathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

• Erdős, Paul; Hajnal, András (1958). “On the structure of set-mappings”. Acta Mathematica Academiae Scien-tiarum Hungaricae 9: 111–131. doi:10.1007/BF02023868. ISSN 0001-5954. MR 0095124

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

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Chapter 74

Extender (set theory)

In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing largecardinal properties. A nonprincipal ultrafilter is the most basic case of an extender.A (κ, λ)-extender can be defined as an elementary embedding of some modelM of ZFC− (ZFC minus the power setaxiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as acollection of ultrafilters, one for each n-tuple drawn from λ.

74.1 Formal definition of an extender

Let κ and λ be cardinals with κ≤λ. Then, a set E = Ea|a ∈ [λ]<ω is called a (κ,λ)-extender if the followingproperties are satisfied:

1. each Ea is a κ-complete nonprincipal ultrafilter on [κ]<ω and furthermore

(a) at least one Ea is not κ+-complete,(b) for each α ∈ κ , at least one Ea contains the set s ∈ [κ]|a| : α ∈ s .

2. (Coherence) The Ea are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).

3. (Normality) If f is such that s ∈ [κ]|a| : f(s) ∈ max s ∈ Ea , then for some b ⊇ a, t ∈ κ|b| :(f πba)(t) ∈ t ∈ Eb .

4. (Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of theultrapowers Ult(V,Ea)).

By coherence, one means that if a and b are finite subsets of λ such that b is a superset of a, then if X is an elementof the ultrafilter Eb and one chooses the right way to project X down to a set of sequences of length |a|, then X isan element of Ea. More formally, for b = α1, . . . , αn , where α1 < · · · < αn < λ , and a = αi1 , . . . , αim, where m≤n and for j≤m the ij are pairwise distinct and at most n, we define the projection πba : ξ1, . . . , ξn 7→ξi1 , . . . , ξim (ξ1 < · · · < ξn) .Then Ea and Eb cohere if

X ∈ Ea ⇔ s : πba(s) ∈ X ∈ Eb

74.2 Defining an extender from an elementary embedding

Given an elementary embedding j:V→M, which maps the set-theoretic universe V into a transitive inner model M,with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines E = Ea|a ∈ [λ]<ω as follows:

82

74.3. REFERENCES 83

fora ∈ [λ]<ω, X ⊆ [κ]<ω : X ∈ Ea ⇔ a ∈ j(X).

One can then show that E has all the properties stated above in the definition and therefore is a (κ,λ)-extender.

74.3 References• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

• Jech, Thomas (2002). Set Theory (3rd ed.). Springer. ISBN 3-540-44085-2.

Chapter 75

Extendible cardinal

In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivatedby reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe ofsets start to look similar, in the sense that each is elementarily embeddable into a later one.

75.1 Definition

For every ordinal η, a cardinal κ is calledη-extendible if for some ordinal λ there is a nontrivial elementary embeddingj of Vκ₊η into Vλ, where κ is the critical point of j, and as usual Vα denotes the αth level of the von Neumannhierarchy. A cardinal κ is called an extendible cardinal if it is η-extendible for every ordinal η (Kanamori 2003).

75.2 Variants and relation to other cardinals

A cardinal κ is called η-C(n)-extendible if there is an elementary embedding j witnessing that κ is η-extendible (that is,j is elementary from Vκ+η to some Vλ with critical point κ) such that furthermore, Vj(κ) is Σn-correct in V. That is,for everyΣn formula φ, φ holds inVj(κ) if and only if φ holds inV. A cardinal κ is said to beC(n)-extendible if it is η-C(n)-extendible for every ordinal η. Every extendible cardinal is C(1)-extendible, but for n≥1, the least C(n)-extendiblecardinal is never C(n+1)-extendible (Bagaria 2011).Vopěnka’s principle implies the existence of extendible cardinals; in fact, Vopěnka’s principle (for definable classes)is equivalent to the existence of C(n)-extendible cardinals for all n (Bagaria 2011). All extendible cardinals aresupercompact cardinals (Kanamori 2003).

75.3 See also

• List of large cardinal properties

• Reinhardt cardinal

75.4 References

• Bagaria, Joan (23 December 2011). "C(n)-cardinals”. Archive for Mathematical Logic 51 (3-4): 213–240.doi:10.1007/s00153-011-0261-8.

• Friedman, Harvey. “Restrictions and Extensions” (PDF).

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

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75.4. REFERENCES 85

• Reinhardt, W. N. (1974), “Remarks on reflection principles, large cardinals, and elementary embeddings.”,Axiomatic set theory, Proc. Sympos. Pure Math., XIII, Part II, Providence, R. I.: Amer. Math. Soc., pp.189–205, MR 0401475

Chapter 76

Extension (predicate logic)

The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfythe predicate. Such a set of tuples is a relation.For example the statement "d2 is the weekday following d1" can be seen as a truth function associating to each tuple(d2, d1) the value true or false. The extension of this truth function is, by convention, the set of all such tuplesassociated with the value true, i.e.(Monday, Sunday), (Tuesday, Monday), (Wednesday, Tuesday), (Thursday, Wednesday), (Friday, Thursday), (Sat-urday, Friday), (Sunday, Saturday)By examining this extension we can conclude that “Tuesday is the weekday following Saturday” (for example) is false.Using set-builder notation, the extension of the n-ary predicate Φ can be written as

(x1, ..., xn) | Φ(x1, ..., xn) .

76.1 Relationship with characteristic function

If the values 0 and 1 in the range of a characteristic function are identified with the values false and true, respectively –making the characteristic function a predicate – , then for all relations R and predicatesΦ the following two statementsare equivalent:

• Φ is the characteristic function of R;

• R is the extension of Φ .

76.2 See also• Extensionality

• Intension

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Chapter 77

Extensionality

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have thesame external properties. It stands in contrast to the concept of intensionality, which is concerned with whether theinternal definitions of objects are the same.

77.1 Example

Consider the two functions f and g mapping from and to natural numbers, defined as follows:

• To find f(n), first add 5 to n, then multiply by 2.

• To find g(n), first multiply n by 2, then add 10.

These functions are extensionally equal; given the same input, both functions always produce the same value. But thedefinitions of the functions are not equal, and in that intensional sense the functions are not the same.Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionallyidentical. For example, suppose that a town has one person named Joe, who is also the oldest person in the town.Then, the two argument predicates “has one person named”, “is the oldest person in” are intensionally distinct, butextensionally equal for “Joe” in that “town” now.

77.2 In mathematics

The extensional definition of function equality, discussed above, is commonly used in mathematics. Sometimesadditional information is attached to a function, such as an explicit codomain, in which case two functions must notonly agree on all values, but must also have the same codomain, in order to be equal.A similar extensional definition is usually employed for relations: two relations are said to be equal if they have thesame extensions.In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements.In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions—withtheir extension as stated above, so that it is impossible for two relations or functions with the same extension to bedistinguished.Other mathematical objects are also constructed in such a way that the intuitive notion of “equality” agrees with set-level extensional equality; thus, equal ordered pairs have equal elements, and elements of a set which are related byan equivalence relation belong to the same equivalence class.Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonlyused to maintain a difference between intensional equality and a more general equivalence relation (which generallyhas poor constructibility or decidability properties).

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88 CHAPTER 77. EXTENSIONALITY

77.3 See also• Duck typing

• Structural typing

• Univalence axiom

77.4 References

Chapter 78

Fiber (mathematics)

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context:

1. In naive set theory, the fiber of the element y in the set Y under a map f : X → Y is the inverse image of thesingleton y under f.

2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because,in general, not every point is closed.

78.1 Definitions

78.1.1 Fiber in naive set theory

Let f : X → Y be a map. The fiber of an element y ∈ Y , commonly denoted by f−1(y) , is defined as

f−1(y) = x ∈ X | f(x) = y.

In various applications, this is also called:

• the inverse image of y under the map f

• the preimage of y under the map f

• the level set of the function f at the point y.

The term level set is only used if f maps into the real numbers and so y is simply a number. If f is a continuousfunction and if y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and moregenerally a hypersurface of dimension d-1.

78.1.2 Fiber in algebraic geometry

In algebraic geometry, if f : X → Y is a morphism of schemes, the fiber of a point p in Y is the fibered productX ×Y Spec k(p) where k(p) is the residue field at p.

78.2 Terminological variance

The recommended practice is to use the terms fiber, inverse image, preimage, and level set as follows:

• the fiber of the element y under the map f

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90 CHAPTER 78. FIBER (MATHEMATICS)

• the inverse image of the set y under the map f

• the preimage of the set y under the map f

• the level set of the function f at the point y.

By abuse of language, the following terminology is sometimes used but should be avoided:

• the fiber of the map f at the element y• the inverse image of the map f at the element y• the preimage of the map f at the element y• the level set of the point y under the map f.

78.3 See also• Fibration

• Fiber bundle

• Fiber product

• Image (category theory)

• Image (mathematics)

• Inverse relation

• Kernel (mathematics)

• Level set

• Preimage

• Relation

• Zero set

Chapter 79

Finite character

In mathematics, a family F of sets is of finite character provided it has the following properties:

1. For each A ∈ F , every finite subset of A belongs to F .

2. If every finite subset of a given set A belongs to F , then A belongs to F .

79.1 Properties

A family F of sets of finite character enjoys the following properties:

1. For each A ∈ F , every (finite or infinite) subset of A belongs to F .

2. Tukey’s lemma: In F , partially ordered by inclusion, the union of every chain of elements of F also belong toF , therefore, by Zorn’s lemma, F contains at least one maximal element.

79.2 Example

Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finitecharacter (because a subset X ⊆ V is linearly dependent iff X has a finite subset which is linearly dependent). There-fore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family isa vector basis, every vector space has a (possibly infinite) vector basis.This article incorporates material from finite character on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

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Chapter 80

Friedberg numbering

In computability theory, a Friedberg numbering is a numbering (enumeration) of the set of all partial recursivefunctions that has no repetitions: each partial recursive function appears exactly once in the enumeration (Vereščaginand Shen 2003:30).The existence of such numberings was established by Richard M. Friedberg in 1958 (Cutland 1980:78).

80.1 References• Nigel Cutland (1980), Computability: An Introduction to Recursive Function Theory, Cambridge UniversityPress. ISBN 9780521294652.

• Richard M. Friedberg (1958), Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set.III. Enumeration Without Duplication, Journal of Symbolic Logic 23:3, pp. 309–316.

• Nikolaj K. Vereščagin and A. Shen (2003), Computable Functions, American Mathematical Soc.

80.2 External links• Institute of Mathematics

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Chapter 81

Gabbay’s separation theorem

In mathematical logic and computer science, Gabbay’s separation theorem, named after Dov Gabbay, states thatany arbitrary temporal logic formula can be rewritten in a logically equivalent “past → future” form. I.e. the futurebecomes what must be satisfied.[1] This form can be used as execution rules; a MetateM program is a set of suchrules.[2]

81.1 References[1] Fisher, Michael David; Gabbay, Dov M.; Vila, Lluis (2005), Handbook of Temporal Reasoning in Artificial Intelligence,

Foundations of Artificial Intelligence 1, Elsevier, p. 150, ISBN 9780080533360.

[2] Kowalski, Robert A.; Sadri, Fariba (1996), “Towards a Unified Agent Architecture That Combines Rationality with Re-activity”, Logic in Databases: International Workshop LID '96, San Miniato, Italy, July 1ÔÇô2, 1996, Proceedings, LectureNotes in Computer Science 1154, Springer-Verlag, pp. 137–149, doi:10.1007/BFb0031739, ISBN 3-540-61814-7.

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Chapter 82

Game-theoretic rough sets

The rough sets can be used to induce three-way classification decisions. The positive negative and boundary regionscan be interpreted as regions of acceptance, rejection and deferment decisions, respectively. The probabilistic roughset model extends the conventional rough sets by providing more effective way for classifying objects. Amain result ofprobabilistic rough sets is the interpretation of three-way decisions using a pair of probabilistic thresholds. The game-theoretic rough set model determine and interprets the required thresholds by utilizing a game-theoretic environmentfor analyzing strategic situations between cooperative or conflicting decision making criteria. The essential idea is toimplement a game for investigating how the probabilistic thresholds may change in order to improve the rough setbased decision making.[1][2][3][4] [5]

82.1 References[1] N. Azam, J. T. Yao, Analyzing Uncertainties of Probabilistic Rough Set Regions with Game-theoretic Rough Sets, Inter-

national Journal of Approximate Reasoning, Vol. 55, No.1, pp 142-155, 2014.

[2] Y. Zhang, Optimizing Gini Coefficient of Probabilistic Rough Set Regions using Game-Theoretic Rough Sets, Proceedingof 26th Annual IEEE Canadian Conference on Electrical and Computer Engineering (CCECE'13), Regina, Canada, May5–8, 2013, pp 699–702

[3] J.P. Herbert, J.T. Yao, Game-theoretic Rough Sets, Fundamenta Informaticae, , 108 (3–4): pp. 267–286, 2011.

[4] J.T. Yao, J.P. Herbert, A Game-Theoretic Perspective on Rough Set Analysis, 2008 International Forum on KnowledgeTechnology (IFKT'08), Chongqing, Journal of Chongqing University of Posts and Telecommunications, Vol. 20, No. 3,pp 291–298, 2008.

[5] Y. Zhang, J.T. Yao, RuleMeasures TradeoffUsingGame-theoretic Rough Sets, Proceeding of the International Conferenceon Brian Informatics (BI'12), Macau, China, Dec 4–7, 2012, Lecture Notes in Computer Science 7670, pp 348–359.

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Chapter 83

Generic filter

In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a tech-nique used for many purposes, but especially to establish the independence of certain propositions from certainformal theories, such as ZFC. For example, Paul Cohen used the method to establish that ZFC, if consistent, cannotprove the continuum hypothesis, which states that there are exactly aleph-one real numbers. In the contemporaryre-interpretation of Cohen’s proof, it proceeds by constructing a generic filter that codes more than ℵ1 reals, withoutchanging the value of ℵ1 .Formally, let P be a poset (partially ordered set), and let F be a filter on P; that is, F is a subset of P such that:

1. F is nonempty

2. If p,q∈P and p≤q and p is an element of F, then q is an element of F (F is closed upward)

3. If p and q are elements of F, then there is an element r of F such that r≤p and r≤q (any two elements of F arecompatible)

Now if D is a collection of dense open subsets of P, in the topology whose basic open sets are all sets of the formq|q≤p for particular p in P, then F is said to be D-generic if F meets all sets in D; that is,

F ∩ E = ∅, for all E ∈ D

Similarly, ifM is a transitive model of ZFC (or some sufficient fragment thereof), with P an element ofM, then F issaid to beM-generic, or sometimes generic overM, if F meets all dense open subsets of P that are elements of M.

83.1 References• K. Ciesielski, Set Theory for the Working Mathematician, London Mathematical Society

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Chapter 84

Ground axiom

In set theory, the ground axiom was introduced by Hamkins (2005) and Reitz (2007). It states that the universe isnot a nontrivial set forcing extension of an inner model.

84.1 References• Hamkins, Joel David (2005), “The Ground Axiom”, Oberwolfach Report 55: 3160–3162

• Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh (2008), “The ground axiom is consistent with V ≠HOD”, Proceedings of the American Mathematical Society 136 (8): 2943–2949, doi:10.1090/S0002-9939-08-09285-X, ISSN 0002-9939, MR 2399062

• Reitz, Jonas (2007), “The ground axiom”, Journal of Symbolic Logic 72 (4): 1299–1317, doi:10.2178/jsl/1203350787,ISSN 0022-4812, MR 2371206

• Jonas Reitz (2008). The Ground Axiom (Ph.D.). CUNY Graduate Center.

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Chapter 85

Hartogs number

In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. Itwas shown by Friedrich Hartogs in 1915, from ZF alone (that is, without using the axiom of choice), that there is aleast well-ordered cardinal greater than a given well-ordered cardinal.To define the Hartogs number of a set it is not in fact necessary that the set be well-orderable: If X is any set, then theHartogs number of X is the least ordinal α such that there is no injection from α into X. If X cannot be well-ordered,then we can no longer say that this α is the least well-ordered cardinal greater than the cardinality of X, but it remainsthe least well-ordered cardinal not less than or equal to the cardinality of X. The map taking X to α is sometimescalled Hartogs’ function.

85.1 Proof

Given some basic theorems of set theory, the proof is simple. Let α = β ∈ Ord | ∃i : β → X . First, we verifythat α is a set.

1. X × X is a set, as can be seen in axiom of power set.

2. The power set of X × X is a set, by the axiom of power set.

3. The class W of all reflexive well-orderings of subsets of X is a definable subclass of the preceding set, so it isa set by the axiom schema of separation.

4. The class of all order types of well-orderings inW is a set by the axiom schema of replacement, as

(Domain(w), w) ∼= (β, ≤)can be described by a simple formula.

But this last set is exactly α.Now because a transitive set of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injectionfrom α into X, then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with noinjection into X. Given β < α, β ∈ α so there is an injection from β into X.

85.2 References• Hartogs, Fritz (1915). "Über das Problem der Wohlordnung”. Mathematische Annalen (in German) 76 (4):438–443. doi:10.1007/BF01458215. JFM 45.0125.01. Available at the DigiZeitschriften.

• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

• Charles Morgan. “Axiomatic set theory” (PDF). Course Notes. University of Bristol. Retrieved 2010-04-10.

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Chapter 86

Herbrand interpretation

In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbolsare assigned very simple meanings.[1] Specifically, every constant is interpreted as itself, and every function symbolis interpreted as the function that applies it. The interpretation also defines predicate symbols as denoting a subset ofthe relevant Herbrand base, effectively specifying which ground atoms are true in the interpretation. This allows thesymbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then thereis a Herbrand interpretation that satisfies them. Moreover, Herbrand’s theorem states that if S is unsatisfiable thenthere is a finite unsatisfiable set of ground instances from the Herbrand universe defined by S. Since this set is finite,its unsatisfiability can be verified in finite time. However there may be an infinite number of such sets to check.It is named after Jacques Herbrand.

86.1 See also• Herbrand structure

• Interpretation (logic)

• Interpretation (model theory)

86.2 Notes[1] Ben Coppin (2004). Artificial Intelligence Illuminated. Jones & Bartlett Learning. p. 231. ISBN 978-0-7637-3230-1.

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Chapter 87

Hereditarily countable set

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductivedefinition is in fact well-founded and can be expressed in the language of first-order set theory. A set is hereditarilycountable if and only if it is countable, and every element of its transitive closure is countable. If the axiom ofcountable choice holds, then a set is hereditarily countable if and only if its transitive closure is countable.The class of all hereditarily countable sets can be proven to be a set from the axioms of Zermelo–Fraenkel set theory(ZF) without any form of the axiom of choice, and this set is designated Hℵ1 . The hereditarily countable sets forma model of Kripke–Platek set theory with the axiom of infinity (KPI), if the axiom of countable choice is assumedin the metatheory.If x ∈ Hℵ1 , then Lω1(x) ⊂ Hℵ1 .More generally, a set is hereditarily of cardinality less than κ if and only if it is of cardinality less than κ, and allits elements are hereditarily of cardinality less than κ; the class of all such sets can also be proven to be a set fromthe axioms of ZF, and is designated Hκ . If the axiom of choice holds and the cardinal κ is regular, then a set ishereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.

87.1 See also• Hereditarily finite set

• Constructible universe

87.2 External links• “On Hereditarily Countable Sets” by Thomas Jech

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Chapter 88

Hereditary set

In set theory, a hereditary set (or pure set) is a set whose elements are all hereditary sets. That is, all elements ofthe set are themselves sets, as are all elements of the elements, and so on.

88.1 Examples

For example, it is vacuously true that the empty set is a hereditary set, and thus the set ∅ containing only the emptyset ∅ is a hereditary set.

88.2 In formulations of set theory

In formulations of set theory that are intended to be interpreted in the von Neumann universe or to express the contentof Zermelo–Fraenkel set theory, all sets are hereditary, because the only sort of object that is even a candidate to bean element of a set is another set. Thus the notion of hereditary set is interesting only in a context in which there maybe urelements.

88.3 Assumptions

The inductive definition of hereditary sets presupposes that set membership is well-founded (i.e., the axiom of reg-ularity), otherwise the recurrence may not have a unique solution. However, it can be restated non-inductively asfollows: a set is hereditary if and only if its transitive closure contains only sets. In this way the concept of hereditarysets can also be extended to non-well-founded set theories in which sets can be members of themselves. For example,a set that contains only itself is a hereditary set.

88.4 See also• Hereditarily countable set

• Well-founded set

88.5 References• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.

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Chapter 89

Heyting arithmetic

In mathematical logic,Heyting arithmetic (sometimes abbreviated HA) is an axiomatization of arithmetic in accor-dance with the philosophy of intuitionism (Troelstra 1973:18). It is named after Arend Heyting, who first proposedit.

89.1 Introduction

Heyting arithmetic adopts the axioms of Peano arithmetic (PA), but uses intuitionistic logic as its rules of inference.In particular, the law of the excluded middle does not hold in general, though the induction axiom can be used toprove many specific cases. For instance, one can prove that ∀ x, y ∈ N : x = y ∨ x ≠ y is a theorem (any two naturalnumbers are either equal to each other, or not equal to each other). In fact, since "=" is the only predicate symbolin Heyting arithmetic, it then follows that, for any quantifier-free formula p, ∀ x, y, z, … ∈ N : p ∨ ¬p is a theorem(where x, y, z… are the free variables in p).

89.2 History

Kurt Gödel studied the relationship between Heyting arithmetic and Peano arithmetic. He used the Gödel–Gentzennegative translation to prove in 1933 that if HA is consistent, then PA is also consistent.

89.3 Related concepts

Heyting arithmetic should not be confused with Heyting algebras, which are the intuitionistic analogue of Booleanalgebras.

89.4 See also

• Harrop formula

• BHK interpretation

89.5 References

• Ulrich Kohlenbach (2008), Applied proof theory, Springer.

• Anne S. Troelstra, ed. (1973),Metamathematical investigation of intuitionistic arithmetic and analysis, Springer,1973.

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102 CHAPTER 89. HEYTING ARITHMETIC

89.6 External links• Stanford Encyclopedia of Philosophy: "Intuitionistic Number Theory" by Joan Moschovakis.

• Fragments of Heyting Arithmetic by Wolfgang Burr

Chapter 90

High (computability)

In computability theory, a Turing degree [X] is high if it is computable in 0′, and the Turing jump [X′] is 0′′, whichis the greatest possible degree in terms of Turing reducibility for the jump of a set which is computable in 0′ (Soare1987:71).Similarly, a degree is high n if its n'th jump is the (n+1)'st jump of 0. Even more generally, a degree d is generalizedhigh n if its n'th jump is the n'th jump of the join of d with 0′.

90.1 See also

Low (computability)

90.2 References

Soare, R. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin,1987. ISBN 3-540-15299-7

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Chapter 91

Hilbert–Bernays provability conditions

In mathematical logic, the Hilbert–Bernays provability conditions, named after David Hilbert and Paul Bernays,are a set of requirements for formalized provability predicates in formal theories of arithmetic (Smith 2007:224).These conditions are used in many proofs of Kurt Gödel's second incompleteness theorem. They are also closelyrelated to axioms of provability logic.

91.1 The conditions

Let T be a formal theory of arithmetic with a formalized provability predicate Prov(n), which is expressed as aformula of T with one free number variable. For each formula φ in the theory, let #(φ) be the Gödel number of φ.The Hilbert–Bernays provability conditions are:

1. If T proves a sentence φ then T proves Prov(#(φ)).

2. For every sentence φ, T proves Prov(#(φ)) → Prov(#(Prov(#(φ))))

3. T proves that Prov(#(φ → ψ)) and Prov(#(φ)) imply Prov (#(ψ))

91.2 References• Smith, Peter (2007). An introduction to Gödel’s incompleteness theorems. Cambridge University Press. ISBN978-0-521-67453-9

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Chapter 92

Homogeneous (large cardinal property)

In set theory and in the context of a large cardinal property, a subset, S, of D is homogeneous for a function f iffor some natural number n, P=n(D) (see Powerset#Subsets of limited cardinality) is the domain of f and for someelement r of the range of f, every member of P=n(S) is mapped to r. That is, f is constant on the unordered n-tuplesof elements of S.

92.1 See also• Ramsey’s theorem

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Chapter 93

Homogeneous tree

In descriptive set theory, a tree over a product set Y ×Z is said to be homogeneous if there is a system of measures⟨µs | s ∈ <ωY ⟩ such that the following conditions hold:

• µs is a countably-additive measure on t | ⟨s, t⟩ ∈ T .

• The measures are in some sense compatible under restriction of sequences: if s1 ⊆ s2 , then µs1(X) =1 ⇐⇒ µs2(t | t lh(s1) ∈ X) = 1 .

• If x is in the projection of T , the ultrapower by ⟨µxn | n ∈ ω⟩ is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:

• There are ⟨µs | s ∈ ωY ⟩ such that if x is in the projection of [T ] and ∀n ∈ ω µxn(Xn) = 1 , then there isf ∈ ωZ such that ∀n ∈ ω f n ∈ Xn . This condition can be thought of as a sort of countable completenesscondition on the system of measures.

T is said to be κ -homogeneous if each µs is κ -complete.Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.

93.1 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-

ican Mathematical Society (Journal of the American Mathematical Society, Vol. 2, No. 1) 2 (1): 71–125.doi:10.2307/1990913. JSTOR 1990913.

106

Chapter 94

Homogeneously Suslin set

In descriptive set theory, a set S is said to be homogeneously Suslin if it is the projection of a homogeneous tree. Sis said to be κ -homogeneously Suslin if it is the projection of a κ -homogeneous tree.If A ⊆ ωω is a 1

1 set and κ is a measurable cardinal, then A is κ -homogeneously Suslin. This result is important inthe proof that the existence of a measurable cardinal implies that 1

1 sets are determined.

94.1 See also• Projective determinacy

94.2 References• Martin, Donald A. and John R. Steel (Jan 1989). “A Proof of Projective Determinacy”. Journal of the Amer-

ican Mathematical Society (American Mathematical Society) 2 (1): 71–125. doi:10.2307/1990913. JSTOR1990913.

107

Chapter 95

Honest leftmost branch

In set theory, an honest leftmost branch of a tree T on ω × γ is a branch (maximal chain) ƒ ∈ [T] such that for eachbranch g ∈ [T], one has ∀ n ∈ ω : ƒ(n) ≤ g(n). Here, [T] denotes the set of branches of maximal length of T, ω is theordinal (represented by the natural numbers N) and γ is some other ordinal.

95.1 See also• scale (computing)

• Suslin set

95.2 References• Akihiro Kanamori, The higher infinite, Perspectives in Mathematical Logic, Springer, Berlin, 1997.

• Yiannis N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980.

108

Chapter 96

Indiscernibles

In mathematical logic, indiscernibles are objects which cannot be distinguished by any property or relation definedby a formula. Usually only first-order formulas are considered.

96.1 Examples

If a, b, and c are distinct and a, b, c is a set of indiscernibles, then, for example, for each binary formula φ, wemust have

[φ(a, b)∧φ(b, a)∧φ(a, c)∧φ(c, a)∧φ(b, c)∧φ(c, b)]∨[¬φ(a, b)∧¬φ(b, a)∧¬φ(a, c)∧¬φ(c, a)∧¬φ(b, c)∧¬φ(c, b)] .

Historically, the identity of indiscernibles was one of the laws of thought of Gottfried Leibniz.

96.2 Generalizations

In some contexts one considers the more general notion of order-indiscernibles, and the term sequence of indis-cernibles often refers implicitly to this weaker notion. In our example of binary formulas, to say that the triple (a, b,c) of distinct elements is a sequence of indiscernibles implies

([φ(a, b)∧φ(a, c)∧φ(b, c)]∨[¬φ(a, b)∧¬φ(a, c)∧¬φ(b, c)])∧([φ(b, a)∧φ(c, a)∧φ(c, b)]∨[¬φ(b, a)∧¬φ(c, a)∧¬φ(c, b)]) .

96.3 Applications

Order-indiscernibles feature prominently in the theory of Ramsey cardinals, Erdős cardinals, and Zero sharp.

96.4 See also• Rough set

96.5 References• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

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Chapter 97

Inductive set

This article is about the notion in descriptive set theory. For the use in foundations of mathematics, see axiom ofinfinity.

Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of Zorn’slemma when nonempty.

In descriptive set theory, an inductive set of real numbers (or more generally, an inductive subset of a Polish space)is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ1n formula, forsome natural number n, together with a real parameter.The inductive sets form a boldface pointclass; that is, they are closed under continuous preimages. In the Wadgehierarchy, they lie above the projective sets and below the sets in L(R). Assuming sufficient determinacy, the class ofinductive sets has the scale property and thus the prewellordering property.

97.1 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

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Chapter 98

Ineffable cardinal

In themathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introducedby Jensen & Kunen (1969).A cardinal number κ is called almost ineffable if for every f : κ → P(κ) (where P(κ) is the powerset of κ )with the property that f(δ) is a subset of δ for all ordinals δ < κ , there is a subset S of κ having cardinal κ andhomogeneous for f , in the sense that for any δ1 < δ2 in S , f(δ1) = f(δ2) ∩ δ1 .A cardinal number κ is called ineffable if for every binary-valued function f : P=2(κ) → 0, 1 , there is a stationarysubset of κ on which f is homogeneous: that is, either f maps all unordered pairs of elements drawn from that subsetto zero, or it maps all such unordered pairs to one.More generally, κ is called n -ineffable (for a positive integer n ) if for every f : P=n(κ) → 0, 1 there is astationary subset of κ on which f is n -homogeneous (takes the same value for all unordered n -tuples drawn fromthe subset). Thus, it is ineffable if and only if it is 2-ineffable.A totally ineffable cardinal is a cardinal that is n -ineffable for every 2 ≤ n < ℵ0 . If κ is (n + 1) -ineffable, thenthe set of n -ineffable cardinals below κ is a stationary subset of κ .Totally ineffable cardinals are of greater consistency strength than subtle cardinals and of lesser consistency strengththan remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.

98.1 References• Friedman, Harvey (2001), “Subtle cardinals and linear orderings”,Annals of Pure and Applied Logic 107 (1–3):1–34, doi:10.1016/S0168-0072(00)00019-1.

• Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V , Unpublished manuscript

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Chapter 99

Infinite descending chain

Given a set S with a partial order ≤, an infinite descending chain is an infinite, strictly decreasing sequence ofelements x1 > x2 > ... > xn > ...

As an example, in the set of integers, the chain −1, −2, −3, ... is an infinite descending chain, but there exists noinfinite descending chain on the natural numbers, as every chain of natural numbers has a minimal element.If a partially ordered set does not possess any infinite descending chains, it is said then, that it satisfies the descendingchain condition. Assuming the axiom of choice, the descending chain condition on a partially ordered set is equiv-alent to requiring that the corresponding strict order is well-founded. A stronger condition, that there be no infinitedescending chains and no infinite antichains, defines the well-quasi-orderings. A totally ordered set without infinitedescending chains is called well-ordered.

99.1 See also• Artinian

99.2 References• Yiannis N. Moschovakis (2006) Notes on set theory, Undergraduate Texts in Mathematics (Birkhäuser) ISBN0-387-28723-X, p.116

112

Chapter 100

Information diagram

Venn diagram for various information measures associated with correlated variables X and Y. The area contained by both circlesis the joint entropy H(X,Y). The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditionalentropy H(X|Y). The circle on the right (blue and violet) is H(Y), with the blue being H(Y|X). The violet is the mutual informationI(X;Y).

An information diagram is a type of Venn diagram used in information theory to illustrate relationships amongShannon’s basic measures of information: entropy, joint entropy, conditional entropy and mutual information.[1][2]Information diagrams are a useful pedagogical tool for teaching and learning about these basic measures of informa-tion, but using such diagrams carries some non-trivial implications. For example, Shannon’s entropy in the context ofan information diagram must be taken as a signed measure. (See the article Information theory and measure theoryfor more information.)

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114 CHAPTER 100. INFORMATION DIAGRAM

Venn diagram of information theoretic measures for three variables x, y, and z. Each circle represents an individual entropy: H(x)is the lower left circle, H(y) the lower right, and H(z) is the upper circle. The intersections of any two circles represents the mutualinformation for the two associated variables (e.g. I(x;z) is yellow and gray). The union of any two circles is the joint entropy for thetwo associated variables (e.g. H(x,y) is everything but green). The joint entropy H(x,y,z) of all three variables is the union of all threecircles. It is partitioned into 7 pieces, red, blue, and green being the conditional entropies H(x|y,z), H(y|x,z), H(z|x,y) respectively,yellow, magenta and cyan being the conditional mutual informations I(x;z|y), I(y;z|x) and I(x;y|z) respectively, and gray being themultivariate mutual information I(x;y;z). The multivariate mutual information is the only one of all that may be negative.

100.1 References[1] Fazlollah Reza. An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994. ISBN

0-486-68210-2

[2] R. W. Yeung, A First Course in Information Theory. Norwell, MA/New York: Kluwer/Plenum, 2002.

Chapter 101

Institutional model theory

This page is about the concept in mathematical logic. For the concept in sociology, see Institutional logic .Institutional model theory generalizes a large portion of first-order model theory to an arbitrary logical system.

101.1 Overview

The notion of “logical system” here is formalized as an institution. Institutions constitute a model-oriented meta-theory on logical systems similar to how the theory of rings and modules constitute a meta-theory for classical linearalgebra. Another analogy can be made with universal algebra versus groups, rings, modules etc. By abstracting awayfrom the realities of the actual conventional logics, it can be noticed that institution theory comes in fact closer to therealities of non-conventional logics.Institutional model theory analyzes and generalizes classical model-theoretic notions and results, like

• Elementary diagrams

• Elementary embeddings

• Ultraproducts, Los’ theorem

• Saturated models

• axiomatizability

• Varieties, Birkhoff axiomatizability

• Craig interpolation

• Robinson consistency

• Beth definability

• Gödel's completeness theorem

For each concept and theorem, the infrastructure and properties required are analyzed and formulated as conditionson institutions, thus providing a detailed insight on which properties of first-order logic they rely and how much theycan be generalized to other logics.

101.2 References

101.3 Further reading• Razvan Diaconescu: Institution-Independent Model Theory. Birkhäuser, 2008. ISBN 978-3-7643-8707-5.

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116 CHAPTER 101. INSTITUTIONAL MODEL THEORY

• Razvan Diaconescu: Jewels of Institution-Independent Model Theory. In: K. Futatsugi, J.-P. Jouannaud, J.Meseguer (eds.): Algebra, Meaning and Computation. Essays Dedicated to Joseph A. Goguen on the Occasionof His 65th Birthday. Lecture Notes in Computer Science 4060, p. 65-98, Springer-Verlag, 2006.

• Marius Petria and Rãzvan Diaconescu: Abstract Beth definability in institutions. Journal of Symbolic Logic71(3), p. 1002-1028, 2006.

• Daniel Gǎinǎ and Andrei Popescu: An institution-independent generalisation of Tarski’s elementary chaintheorem, Journal of Logic and Computation 16(6), p. 713-735, 2006.

• Till Mossakowski, Joseph Goguen, Rãzvan Diaconescu, Andrzej Tarlecki: What is a Logic?. In Jean-YvesBeziau, editor, Logica Universalis, pages 113-133. Birkhauser, 2005.

• Andrzej Tarlecki: Quasi-varieties in abstract algebraic institutions. Journal of Computer and System Sciences33(3), p. 333-360, 1986.

101.4 External links• Razvan Diaconescu’s publication list - contains recent work on institutional model theory

Chapter 102

Iterable cardinal

In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman (2011) and studied by Gitmanand Welch (2011). Gitman defined a cardinal κ to be α-iterable if every subset of κ is contained in a weak κ-modelM for which there exists an M-ultrafilter on κ allowing an iteration of length α.

102.1 References• Gitman, Victoria (2011), “Ramsey-like cardinals I”, J. Symbolic Logic 76 (2): 519–540, doi:10.2178/jsl/1305810762,MR 2830435

• Gitman, Victoria; Welch, P. D. (2011), “Ramsey-like cardinals II”, J. Symbolic Logic 76 (2): 541–560,doi:10.2178/jsl/1305810763, MR 2830435

102.2 External links• Diagram of iterable cardinals

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Chapter 103

Jensen’s covering theorem

In set theory, Jensen’s covering theorem states that if 0# does not exist then every uncountable set of ordinals iscontained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universeis close to the universe of all sets. The first proof appeared in (Devlin & Jensen 1975). Silver later gave a fine structurefree proof using his machines and finally Magidor (1990) gave an even simpler proof.The converse of Jensen’s covering theorem is also true: if 0# exists then the countable set of all cardinals less thanℵω cannot be covered by a constructible set of cardinality less than ℵω.In his book Proper Forcing, Shelah proved a strong form of Jensen’s covering lemma.

103.1 References• Devlin, Keith I.; Jensen, R. Björn (1975), “Marginalia to a theorem of Silver”, ISILC Logic Conference (Proc.

Internat. Summer Inst. and Logic Colloq., Kiel, 1974), Lecture notes in mathematics 499, Berlin, New York:Springer-Verlag, pp. 115–142, doi:10.1007/BFb0079419, ISBN 978-3-540-07534-9, MR 0480036

• Magidor, Menachem (1990), “Representing sets of ordinals as countable unions of sets in the core model”,Transactions of the American Mathematical Society 317 (1): 91–126, doi:10.2307/2001455, ISSN 0002-9947,MR 939805

• Mitchell,William (2010), “The covering lemma”,Handbook of Set Theory, Springer, pp. 1497–1594, doi:10.1007/978-1-4020-5764-9_19, ISBN 978-1-4020-4843-2

• Shelah, Saharon (1982), Proper forcing, Lecture Notes in Mathematics 940, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0096536, ISBN 978-3-540-11593-9, MR 675955

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Chapter 104

Joint embedding property

In universal algebra and model theory, a class of structures K is said to have the joint embedding property if for allstructures A and B in K, there is a structure C in K such that both A and B have embeddings into C.It is one of the three properties used to define the age of a structure.A similar but different notion to the joint embedding property is the amalgamation property. To see the difference,first consider the class K (or simply the set) containing three models with linear orders, L1 of size one, L2 of size two,and L3 of size three. This class K has the joint embedding property because all three models can be embedded intoL3. However, K does not have the amalgamation property. The counterexample for this starts with L1 containing asingle element e and extends in two different ways to L3, one in which e is the smallest and the other in which e is thelargest. Now any common model with an embedding from these two extensions must be at least of size five so thatthere are two elements on either side of e.Now consider the class of algebraically closed fields. This class has the amalgamation property since any two fieldextensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embeddedinto a common field when the characteristic of the fields differ.

104.1 References• Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 0-521-58713-1.

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Chapter 105

Judgment (mathematical logic)

For other uses, see Judgment (disambiguation).

In mathematical logic, a judgment can be an assertion about occurrence of a free variable in an expression of theobject language, or about provability of a proposition (either as a tautology or from a given context), but judgmentscan be also other inductively definable assertions in the metatheory. Judgments are used for example in formalizingdeduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequenceof judgments, and their conclusion is a judgment as well. Also the result of a proof expresses a judgment, and theused hypotheses are formed as a sequence of judgments.A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in anyof their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules.Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalizethe Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simpleform. The same cannot be done with the other two deductions systems: as context is changed in some of their rulesof inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want touse them just for proving derivability of tautologies.This basic diversity among the various calculi allows such difference, that the same basic thought (e.g. deductiontheorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly as arule of inference in natural deduction.In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the twofields, e.g. Curry-Howard correspondence). The abstraction in the notion of judgment in mathematical logic can beexploited also in foundation of type theory as well.

105.1 See also• Simply typed lambda calculus

• Mathematical logic

105.2 External links• “Judgments in formal systems”. Everything2.

• Pfenning, Frank (Spring 2004). “Natural Deduction” (PDF). 15-815 Automated Theorem Proving.

• Martin-Löf, Per (1983). “On the meaning of the logical constants and the justifications of the logical laws”.Siena Lectures.

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Chapter 106

Jónsson cardinal

In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number.An uncountable cardinal number κ is said to be Jónsson if for every function f: [κ]<ω → κ there is a set H of ordertype κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ.Every Rowbottom cardinal is Jónsson. By a theorem of Eugene M. Kleinberg, the theories ZFC + “there is aRowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent. William Mitchell proved, withthe help of the Dodd-Jensen core model that the consistency of the existence of a Jónsson cardinal implies the con-sistency of the existence of a Ramsey cardinal.In general, Jónsson cardinals need not be large cardinals in the usual sense: they can be singular. But the existenceof a singular Jónsson cardinal is equiconsistent to the existence of a measurable cardinal. Using the axiom of choice,a lot of small cardinals (the ℵn , for instance) can be proved to be not Jónsson. Results like this need the axiom ofchoice, however: The axiom of determinacy does imply that for every positive natural number n, the cardinal ℵn isJónsson.A Jónsson algebra is an algebra with no proper subalgebras of the same cardinality. (They are unrelated to Jónsson–Tarski algebras). Here an algebra means a model for a language with a countable number of function symbols, inother words a set with a countable number of functions from finite products of the set to itself. A cardinal is a Jónssoncardinal if and only if there are no Jónsson algebras of that cardinality. The existence of Jónsson functions shows thatif algebras are allowed to have infinitary operations, then there are no analogues of Jónsson cardinals.

106.1 References• Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd eded.). Springer. ISBN 3-540-00384-3.

• Jónsson, Bjarni (1972), Topics in universal algebra, Lecture Notes in Mathematics 250, Berlin, New York:Springer-Verlag, doi:10.1007/BFb0058648, MR 0345895

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Chapter 107

Kanamori–McAloon theorem

In mathematical logic, the Kanamori–McAloon theorem, due to Kanamori & McAloon (1987), gives an exampleof an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certainfinitistic special case of a theorem in Ramsey theory due to Erdős and Rado is not provable in Peano arithmetic.

107.1 See also• Paris–Harrington theorem

• Goodstein’s theorem

• Kruskal’s tree theorem

107.2 References• Kanamori, Akihiro; McAloon, Kenneth (1987), “On Gödel incompleteness and finite combinatorics”, Annals

of Pure and Applied Logic 33 (1): 23–41, doi:10.1016/0168-0072(87)90074-1, ISSN 0168-0072, MR 870685

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Chapter 108

Kleene–Rosser paradox

In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic areinconsistent, in particular the version of Curry's combinatory logic introduced in 1930, and Church's original lambdacalculus, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibitedby Stephen Kleene and J. B. Rosser in 1935.

108.1 The paradox

Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total,definable number-theoretic functions, which enabled them to construct a term that essentially replicates the Richardparadox in formal language.Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox,and used this to construct a much simpler paradox, now known as Curry’s paradox.

108.2 See also• List of paradoxes

108.3 References• Andrea Cantini, "The inconsistency of certain formal logics", in the Paradoxes and Contemporary Logic entryof Stanford Encyclopedia of Philosophy (2007).

• Kleene, S. C. & Rosser, J. B. (1935). “The inconsistency of certain formal logics”. Annals of Mathematics 36(3): 630–636. doi:10.2307/1968646.

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Chapter 109

Knaster’s condition

In mathematics, a partially ordered set P is said to have Knaster’s condition upwards (sometimes property (K))if any uncountable subset A of P has an upwards-linked uncountable subset. An analogous definition applies toKnaster’s condition downwards.The property is named after Polish mathematician Bronisław Knaster.Knaster’s condition implies a countable chain condition (ccc), and it is sometimes used in conjunction with a weakerform of Martin’s axiom, where the ccc requirement is replaced with Knaster’s condition. Not unlike ccc, Knaster’scondition is also sometimes used as a property of a topological space, in which case it means that the topology (as in,the family of all open sets) with inclusion satisfies the condition.Furthermore, assuming MA( ω1 ), ccc implies Knaster’s condition, making the two equivalent.

109.1 References• Fremlin, David H. (1984). Consequences of Martin’s axiom. Cambridge tracts in mathematics, no. 84. Cam-bridge: Cambridge University Press. ISBN 0-521-25091-9.

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Chapter 110

Kunen’s inconsistency theorem

In set theory, a branch of mathematics, Kunen’s inconsistency theorem, proved by Kenneth Kunen (1971), showsthat several plausible large cardinal axioms are inconsistent with the axiom of choice.Some consequences of Kunen’s theorem are:

• There is no non-trivial elementary embedding of the universeV into itself. In other words, there is no Reinhardtcardinal.

• If j is an elementary embedding of the universe V into an inner modelM, and λ is the smallest fixed point of jabove the critical point κ of j, then M does not contain the set j "λ (the image of j restricted to λ).

• There is no ω-huge cardinal.

• There is no non-trivial elementary embedding of Vλ₊₂ into itself.

It is not known if Kunen’s theorem still holds in ZF (ZFC without the axiom of choice), though Suzuki (1999) showedthat there is no definable elementary embedding from V into V. That is there is no formula J in the language of settheory such that for some parameter p∈V for all sets x∈V and y∈V : j(x) = y ↔ J(x, y, p) .

Kunen used Morse–Kelley set theory in his proof. If the proof is re-written to use ZFC, then one must add theassumption that replacement holds for formulas involving j. Otherwise one could not even show that j "λ exists as aset. The forbidden set j "λ is crucial to the proof. The proof first shows that it cannot be inM. The other parts of thetheorem are derived from that.It is possible to have models of set theory that have elementary embeddings into themselves, at least if one as-sumes some mild large cardinal axioms. For example, if 0# exists then there is an elementary embedding from theconstructible universe L into itself. This does not contradict Kunen’s theorem because if 0# exists then L cannot bethe whole universe of sets.

110.1 See also• Rank-into-rank

110.2 References• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.), Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-88867-3, ISBN 978-3-540-00384-7

• Kunen, Kenneth (1971), “Elementary embeddings and infinitary combinatorics”, J. Symbolic Logic 36 (3):407–413, doi:10.2307/2269948, JSTOR 2269948, MR 0311478

• Suzuki, Akira (1999), “No elementary embedding from V into V is definable from parameters”, The Journalof Symbolic Logic 64 (4): 1591–1594, doi:10.2307/2586799, ISSN 0022-4812, MR 1780073

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126 CHAPTER 110. KUNEN’S INCONSISTENCY THEOREM

• Zapletal, Jindřich (1996), “A new proof of Kunen’s inconsistency”, Proceedings of the American MathematicalSociety 124 (7): 2203–2204, doi:10.1090/S0002-9939-96-03281-9, ISSN 0002-9939, MR 1317054

Chapter 111

Kuratowski’s free set theorem

Kuratowski’s free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics.It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving severallattice theory problems, such as the congruence lattice problem.Denote by [X]<ω the set of all finite subsets of a setX . Likewise, for a positive integer n , denote by [X]n the set ofall n -elements subsets of X . For a mapping Φ: [X]n → [X]<ω , we say that a subset U of X is free (with respectto Φ ), if for any n -element subset V of U and any u ∈ U \ V , u /∈ Φ(V ) ,. Kuratowski published in 1951 thefollowing result, which characterizes the infinite cardinals of the form ℵn .The theorem states the following. Let n be a positive integer and letX be a set. Then the cardinality ofX is greaterthan or equal to ℵn if and only if for every mapping Φ from [X]n to [X]<ω , there exists an (n + 1) -element freesubset of X with respect to Φ .For n = 1 , Kuratowski’s free set theorem is superseded by Hajnal’s set mapping theorem.

111.1 References• P. Erdős, A. Hajnal, A. Máté, R. Rado: Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland, 1984, pp. 282–285.

• C. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14–17.

• John C. Simms (1991) “Sierpiński’s theorem”, Simon Stevin 65: 69–163.

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Chapter 112

Kurepa tree

In set theory, a Kurepa tree is a tree (T, <) of height ω1 , each of whose levels is at most countable, and has atleast ℵ2 many branches. This concept was introduced by Kurepa (1935). The existence of a Kurepa tree (knownas the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axiomsof ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe (Jech1971). More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in theconstructible universe. On the other hand, Silver (1971) showed that if a strongly inaccessible cardinal is Lévycollapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal isin fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then thecardinal ω2 is inaccessible in the constructible universe.A Kurepa tree with fewer than 2ℵ1 branches is known as a Jech–Kunen tree.More generally if κ is an infinite cardinal, then a κ-Kurepa tree is a tree of height κ with more than κ branches but atmost |α| elements of each infinite level α<κ, and the Kurepa hypothesis for κ is the statement that there is a κ-Kurepatree. Sometimes the tree is also assumed to be binary. The existence of a binary κ-Kurepa tree is equivalent to theexistence of aKurepa family: a set of more than κ subsets of κ such that their intersections with any infinite ordinalα<κ form a set of cardinality at most α. The Kurepa hypothesis is false if κ is an ineffable cardinal, and converselyJensen showed that in the constructible universe for any uncountable regular cardinal κ there is a κ-Kurepa tree unlessκ is ineffable.

112.1 Specializing a Kurepa tree

A Kurepa tree can be killed by forcing the existence of a function whose value on any non-root node is an ordinalless than the rank of the node, such that whenever three nodes, one of which is a lower bound for the other two,are mapped to the same ordinal, then the three nodes are comparable. This can be done without collapsing ℵ1 , andresults in a tree with exactly ℵ1 branches.

112.2 See also

• Aronszajn tree

• Suslin tree

112.3 References

• Jech, Thomas J. (1971), “Trees”, J. Symbolic Logic 36: 1–14, doi:10.2307/2271510, JSTOR 2271510, MR0284331, Zbl 0245.02054

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

128

112.3. REFERENCES 129

• Kurepa, G. (1935), “Ensembles ordonnés et ramifiés”, Publ. math. Univ. Belgrade 4: 1–138, JFM 61.0980.01,Zbl 0014.39401

• Silver, Jack (1971), “The independence of Kurepa’s conjecture and two-cardinal conjectures in model theory”,Axiomatic Set Theory, Proc. Sympos. Pure Math. XIII, Providence, R.I.: Amer. Math. Soc., pp. 383–390,MR 0277379, Zbl 0255.02068

Chapter 113

Language equation

Language equations are mathematical statements that resemble numerical equations, but the variables assume valuesof formal languages rather than numbers. Instead of arithmetic operations in numerical equations, the variables arejoined by language operations. Among the most common operations on two languages A and B are the set union A ∪B, the set intersection A ∩ B, and the concatenation A⋅B. Finally, as an operation taking a single operand, the set A*

denotes the Kleene star of the language A. Therefore language equations can be used to represent formal grammars,since the languages generated by the grammar must be the solution of a system of language equations.

113.1 See also• Boolean grammar

• Arden’s Rule

• Set constraint

113.2 External links• Workshop on Theory and Applications of Language Equations (TALE 2007)

• Language equations

130

Chapter 114

Laver function

In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connectedwith supercompact cardinals.

114.1 Definition

If κ is a supercompact cardinal, a Laver function is a function ƒ:κ → Vκ such that for every set x and every cardinalλ ≥ |TC(x)| + κ there is a supercompact measure U on [λ]<κ such that if jU is the associated elementary embeddingthen jU(ƒ)(κ) = x. (Here Vκ denotes the κ-th level of the cumulative hierarchy, TC(x) is the transitive closure of x)

114.2 Applications

The original application of Laver functions was the following theorem of Laver. If κ is supercompact, there is a κ-c.c.forcing notion (P, ≤) such after forcing with (P, ≤) the following holds: κ is supercompact and remains supercompactafter forcing with any κ-directed closed forcing.There are many other applications, for example the proof of the consistency of the proper forcing axiom.

114.3 References• Laver, Richard (1978). “Making the supercompactness of κ indestructible under κ-directed closed forcing”.

Israel Journal of Mathematics 29: 385–388. doi:10.1007/bf02761175. Zbl 0381.03039.

131

Chapter 115

Least fixed point

The function f(x)=x²−4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2-√17/2.

In order theory, a branch of mathematics, the least fixed point (lfp or LFP, sometimes also smallest fixed point) ofa function from a partially ordered set to itself is the fixed point which is less than each other fixed point, accordingto the set’s order. A function need not have a least fixed point, and cannot have more than one.For example, with the usual order on the real numbers, the least fixed point of the real function f(x) = x² is x = 0(since the only other fixed point is 1 and 0 < 1). In contrast, f(x) = x+1 has no fixed point at all, let alone a least one,and f(x)=x has infinitely many fixed points, but no least one.

132

115.1. APPLICATIONS 133

115.1 Applications

Many fixed-point theorems yield algorithms for locating the least fixed point. Least fixed points often have desirableproperties that arbitrary fixed points do not.In mathematical logic and computer science, the least fixed point is related tomaking recursive definitions (see domaintheory and/or denotational semantics for details).Immerman [1][2] and Vardi [3] independently showed the descriptive complexity result that the polynomial-time com-putable properties of linearly ordered structures are definable in FO(LFP), i.e. in first-order logic with a least fixedpoint operator. However, FO(LFP) is too weak to express all polynomial-time properties of unordered structures(for instance that a structure has even size).

115.2 Examples

Let G=(V,A) be a directed graph and v be a vertex. The set of states accessible from v can be defined as the set Swhich is the least fixed-point for the property: v belongs to S and if w belongs to S and there is an arrow from w tox, then x belongs to S. The set of states which are co-accessible from v is defined by a similar least fix-point. On theone hand the strongly connected component of v is the intersection of those two least fixed-point.Let G be a proper context-free grammar. The set E of symbols which produces the emptyword is defined as the leastfixed-point which contains the symbols S such that S → ε , or such that S → S0, . . . , Sn where all symbols Si

belongs to E.

115.3 Greatest fixed points

Greatest fixed points can also be determined, but they are less commonly used than least fixed points. However, incomputer science they, analogously to the least fixed point, give rise to corecursion and codata.

115.4 See also• Fixed point

• Kleene fixed-point theorem

• Knaster–Tarski theorem

115.5 Notes[1] N. Immerman, Relational queries computable in polynomial time, Information and Control 68 (1–3) (1986) 86–104.

[2] Immerman, Neil (1982). “Relational Queries Computable in Polynomial Time”. STOC '82: Proceedings of the fourteenthannual ACM symposium on Theory of computing. pp. 147–152. doi:10.1145/800070.802187. Revised version in Infor-mation and Control, 68 (1986), 86–104.

[3] Vardi, Moshe Y. (1982). “The Complexity of Relational Query Languages”. STOC '82: Proceedings of the fourteenthannual ACM symposium on Theory of computing. pp. 137–146. doi:10.1145/800070.802186.

115.6 References• Immerman, Neil. Descriptive Complexity, 1999, Springer-Verlag.

• Libkin, Leonid. Elements of Finite Model Theory, 2004, Springer.

Chapter 116

LEGO (proof assistant)

LEGO is a proof assistant developed by Randy Pollack at the University of Edinburgh. It implements several typetheories: the Edinburgh Logical Framework (LF), the Calculus of Constructions (CC), the Generalized Calculus ofConstructions (GCC) and the Unified Theory of Dependent Types (UTT).

116.1 External links• Official website

134

Chapter 117

Lightface analytic game

In descriptive set theory, a lightface analytic game is a game whose payoff set A is a Σ11 subset of Baire space; that

is, there is a tree T on ω × ω which is a computable subset of (ω × ω)<ω , such that A is the projection of the set ofall branches of T.The determinacy of all lightface analytic games is equivalent to the existence of 0#.

135

Chapter 118

Limitation of size

In the philosophy of mathematics, specifically the philosophical foundations of set theory, limitation of size is aconcept developed by Philip Jourdain and/or Georg Cantor to avoid Cantor’s paradox. It identifies certain “inconsis-tent multiplicities”, in Cantor’s terminology, that cannot be sets because they are “too large”. In modern terminologythese are called proper classes.

118.1 Use

The axiom of limitation of size is an axiom in some versions of von Neumann–Bernays–Gödel set theory or Morse–Kelley set theory. This axiom says that any class which is not “too large” is a set, and a set cannot be “too large”.“Too large” is defined as being large enough that the class of all sets can be mapped one-to-one into it.

118.2 References• Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. Oxford University Press. ISBN 0-19-853283-0.

136

Chapter 119

Limited principle of omniscience

In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle ofomniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle(Bridges and Richman 1987). The LPO and LLPO axioms are used to gauge the amount of nonconstructivity requiredfor an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the senseof Brouwer.

119.1 Definitions

The limited principle of omniscience states (Bridges and Richman 1987:3):

LPO: For any sequence a0, a1, ... such that each ai is either 0 or 1, the following holds: either ai = 0for all i, or there is a k with ak = 1.

The lesser limited principle of omniscience states:

LLPO: For any sequence a0, a1, ... such that each ai is either 0 or 1, and such that at most one ai isnonzero, the following holds: either a₂i = 0 for all i, or a₂i₊₁ = 0 for all i, where a₂i and a₂i₊₁ are entrieswith even and odd index respectively.

It can be proved constructively that the law of the excluded middle implies LPO, and LPO implies LLPO. However,none of these implications can be reversed in typical systems of constructive mathematics.The term “omniscience” comes from a thought experiment regarding how a mathematician might tell which of thetwo cases in the conclusion of LPO holds for a given sequence (ai). Answering the question “is there a k with ak =1?" negatively, assuming the answer is negative, seems to require surveying the entire sequence. Because this wouldrequire the examination of infinitely many terms, the axiom stating it is possible to make this determination wasdubbed an “omniscience principle” by Bishop (1967).

119.2 References• Bishop, Errett Foundations of Constructive Analysis, 1967. ISBN 4-87187-714-0

• Douglas Bridges and Fred Richman, Varieties of Constructive Mathematics, LondonMathematical Society Lec-ture Notes v. 57, 1987. ISBN 0-521-31802-5

119.3 External links• Constructive mathematics", Douglas Bridges, Stanford Encyclopedia of Philosophy

137

Chapter 120

Lindström’s theorem

In mathematical logic, Lindström’s theorem (named after Swedish logician Per Lindström, who published it in1969) states that first-order logic is the strongest logic [1] (satisfying certain conditions, e.g. closure under classicalnegation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.[2]

Lindström’s theorem is perhaps the best known result of what later became known as abstract model theory,[3] thebasic notion of which is an abstract logic;[4] the more general notion of an institution was later introduced, whichadvances from a set-theoretical notion of model to a category theoretical one.[5] Lindström had previously obtaineda similar result in studying first-order logics extended with Lindström quantifiers.[6]

Lindström’s theorem has been extended to various other systems of logic in particular modal logics by Johan vanBenthem and Sebastian Enqvist.

120.1 Notes[1] In the sense of Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors,

Model-theoretic logics, 1985 ISBN 0-387-90936-2 page 43

[2] A companion to philosophical logic by Dale Jacquette 2005 ISBN 1-4051-4575-7 page 329

[3] Chen Chung Chang; H. Jerome Keisler (1990). Model theory. Elsevier. p. 127. ISBN 978-0-444-88054-3.

[4] Jean-Yves Béziau (2005). Logica universalis: towards a general theory of logic. Birkhäuser. p. 20. ISBN 978-3-7643-7259-0.

[5] Dov M. Gabbay, ed. (1994). What is a logical system?. Clarendon Press. p. 380. ISBN 978-0-19-853859-2.

[6] Jouko Väänänen, Lindström’s Theorem

120.2 References• Per Lindström, “OnExtensions of Elementary Logic”, Theoria 35, 1969, 1–11. doi:10.1111/j.1755-2567.1969.tb00356.x• Johan vanBenthem, “ANewModal LindströmTheorem”, LogicaUniversalis 1, 2007, 125–128. doi:10.1007/s11787-006-0006-3

• Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994),Mathematical Logic (2nd ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94258-2

• Sebastian Enqvist, “A General Lindström Theorem for Some Normal Modal Logics”, Logica Universalis 7,2013, 233–264. doi:10.1007/s11787-013-0078-9

• Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1

• Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and com-plexity, Oxford University Press, 2004, ISBN 0-19-852981-3, section 9.4

138

Chapter 121

Linked set

In mathematics, an upwards linked set A is a subset of a partially ordered set, P, in which any two of elements Ahave a common upper bound in P. Similarly, every pair of elements of a downwards linked set has a lower bound.Note that every centered set is linked, which includes, in particular, every directed set.

121.1 References• Fremlin, David H. (1984). Consequences of Martin’s axiom. Cambridge tracts in mathematics, no. 84. Cam-bridge: Cambridge University Press. ISBN 0-521-25091-9.

139

Chapter 122

LOGCFL

In computational complexity theory, LOGCFL is the complexity class that contains all decision problems that canbe reduced in logarithmic space to a context-free language. This class is situated between NL and AC1, in the sensethat it contains the former and is contained in the latter. Problems that are complete for LOGCFL include manyproblems whose instances can be characterized by acyclic hypergraphs:

• evaluating acyclic Boolean conjunctive queries

• checking the existence of a homomorphism between two acyclic relational structures

• checking the existence of solutions of acyclic constraint satisfaction problems

122.1 See also• List of complexity classes

122.2 External links• Complexity Zoo: LOGCFL

140

Chapter 123

Logic for Computable Functions

See also: Logic of Computable Functions

Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at the universitiesof Edinburgh and Stanford by Robin Milner and others in 1972. LCF introduced the general-purpose programminglanguage ML to allow users to write theorem-proving tactics. Theorems in the system are propositions of a special“theorem” abstract datatype. The ML type system ensures that theorems are derived using only the inference rulesgiven by the operations of the abstract type.Successors include HOL (Higher Order Logic) and Isabelle.

123.1 References• Gordon, Michael J. C. (1996). “From LCF to HOL: a short history”. Retrieved 2007-10-11.

• Milner, Robin (May 1972). Logic for Computable Functions: description of a machine implementation. (PDF).Stanford University.

141

Chapter 124

Logical assertion

In mathematical logic, logical assertion is a statement that asserts that a certain premise is true, and is useful forstatements in proof. It is equivalent to a sequent with an empty antecedent.For example, if p = "x is even”, the implication

(⊢ p) → (x (mod 2) ≡ 0)

is thus true. We can also write this using the logical assertion symbol, as

⊢ ((⊢ p) → (x (mod 2) ≡ 0))

In computer programming and programming language semantics, these are used in the form of assertions; one ex-ample is a loop invariant.

142

Chapter 125

Logical graph

A logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax thatCharles Sanders Peirce developed for logic.In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of agraphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.In the century since Peirce initiated this line of development, a variety of formal systems have branched out fromwhat is abstractly the same formal base of graph-theoretic structures.

125.1 See also• Charles Sanders Peirce bibliography

• Conceptual graph

• Propositional calculus

• Truth table

125.2 External links

Media related to Logical graphs at Wikimedia Commons

• Logical Graph @ Commens Dictionary of Peirce’s Terms

• Existential Graphs, Jay Zeman, ed., U. of Florida. With 4 works by Peirce.

• Frithjof Dau’s page of readings and links on existential graphs includes lists of: books exclusively on existentialgraphs; books containing existential graphs; articles; and some links and downloadables.

• The literature of C.S. Peirce’s Existential Graphs (via Internet Archive), Xin-Wen Liu, Institute of Philosophy,Chinese Academy of Social Sciences, Beijing, PRC. A whole lot there.

143

Chapter 126

Logical machine

Logical machine is a term used by AllanMarquand (1853-1924) in 1883, perhaps in response to the ideas of CharlesSanders Peirce's “Logical Machines” as appearing for example in The American Journal of Psychology, 1. Nov. 1887,p. 165-170 (Google Books Eprint page 165).

126.1 Bibliography• Marquand, Allan

• (1883), “AMachine for Producing Syllogistic Variation” in C. S. Peirce, ed., Studies in Logic, pp. 12–15,along with “Note on an Eight-Term Logical Machine”, p. 16. Google Books Eprint. Book reprinted1983 with introduction by Max Fisch.

• (1886), “A New Logical Machine”, Proceedings of the American Academy of Arts and Sciences 21: 303–07. Google Books Eprint.

• Peirce, C. S.

• (1886 letter), Letter, Peirce to A. Marquand, 1886 December 30, published 1993 in Kloesel, C. et al.,eds., Writings of Charles S. Peirce: A Chronological Edition, Vol. 5. Indiana Univ. Press, pp. 421–3.Google Books Preview.

• (1887), “Logical Machines”, The American Journal of Psychology v. 1, n. 1, Baltimore: N. Murray, pp.165–70. Google Books Eprint. Reprinted in (1976) The New Elements of Mathematics v. III, pt. 1, pp.625–32; (1997) Modern Logic 7:71–77, Project Euclid Eprint; and (2000) Writings of Charles S. Peircev. 6, pp. 65–73.

• Baldwin, Mark James (1902), “Logical Machine”,Dictionary of Philosophy and Psychology, pp. 28–30 GoogleBooks Eprint. Classics in the History of Psychology Eprint.

• Ketner, Kenneth Laine (1984), “The early history of computer design: Charles Sanders Peirce and Marquand’slogical machines”, with the assistance of Arthur Franklin Stewart, Princeton University Library Chronicle, v.45, n. 3, pp. 186–211. PULC 15MB PDF Eprint.

• Dalakov, Georgi (undated), “Charles Peirce and Allan Marquand”, History of Computers and Computing.Eprint.

126.2 See also• Logics for computability

144

Chapter 127

Low (computability)

In computability theory, a Turing degree [X] is low if the Turing jump [X′] is 0′. A set is low if it has low degree.Since every set is computable from its jump, any low set is computable in 0′, but the jump of sets computable in 0′can bound any degree r.e. in 0′ (Schoenfield Jump Inversion). X being low says that its jump X′ has the least possibledegree in terms of Turing reducibility for the jump of a set.A degree is low n if its n'th jump is the n'th jump of 0. A set X is generalized low if it satisfies X′ ≡T X + 0′, that is:if its jump has the lowest degree possible. And a degree d is generalized low n if its n'th jump is the (n-1)'st jumpof the join of d with 0′. More generally, properties of sets which describe their being computationally weak (whenused as a Turing oracle) are referred to under the umbrella term lowness properties.By the Low basis theorem of Jockusch and Soare, any nonempty Π0

1 class in 2ω contains a set of low degree. Thisimplies that, although low sets are computationally weak, they can still accomplish such feats as computing a com-pletion of Peano Arithmetic. In practice, this allows a restriction on the computational power of objects needed forrecursion theoretic constructions: for example, those used in the analyzing the proof-theoretic strength of Ramsey’stheorem.

127.1 See also• High (computability)

• Low Basis Theorem

127.2 References• Soare, Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and com-

putably generated sets. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. ISBN 3-540-15299-7.Zbl 0667.03030.

• Nies, André (2009). Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford UniversityPress. ISBN 978-0-19-923076-1. Zbl 1169.03034.

145

Chapter 128

Low basis theorem

The low basis theorem in computability theory states that every nonemptyΠ01 class in 2ω (see arithmetical hierarchy)

contains a set of low degree (Soare 1987:109). It was first proved by Carl Jockusch and Robert I. Soare in 1972 (Nies2009:57). Cenzer (1993:53) describes it as “perhaps the most cited result in the theory of Π0

1 classes”. The proofuses the method of forcing with Π0

1 classes (Cooper 2004:330).

128.1 References• Cenzer, Douglas (1999). " Π0

1 classes in computability theory”. In Griffor, Edward R. Handbook of com-putability theory. Stud. Logic Found. Math. 140. North-Holland. pp. 37–85. ISBN 0-444-89882-4. MR1720779. Zbl 0939.03047. Theorem 3.6, p. 54.

• Cooper, S. Barry (2004). Computability Theory. Chapman and Hall/CRC. ISBN 1-58488-237-9..

• Jockusch, Carl G., Jr.; Soare, Robert I. (1972). "Π(0, 1) Classes and Degrees of Theories”. Transactionsof the American Mathematical Society 173: 33–56. doi:10.1090/s0002-9947-1972-0316227-0. ISSN 0002-9947. JSTOR 1996261. Zbl 0262.02041. The original publication, including additional clarifying prose.

• Nies, André (2009). Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford UniversityPress. ISBN 978-0-19-923076-1. Zbl 1169.03034. Theorem 1.8.37.

• Soare, Robert I. (1987). Recursively enumerable sets and degrees. A study of computable functions and com-putably generated sets. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. ISBN 3-540-15299-7.Zbl 0667.03030.

146

Chapter 129

Lusin’s separation theorem

This article is about the separation theorem. For the theorem on continuous functions, see Lusin’s theorem.

In descriptive set theory and mathematical logic, Lusin’s separation theorem states that if A and B are disjointanalytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅.[1] It is namedafter Nikolai Luzin, who proved it in 1927.[2]

The theorem can be generalized to show that for each sequence (An) of disjoint analytic sets there is a sequence (Bn)of disjoint Borel sets such that An ⊆ Bn for each n. [1]

An immediate consequence is Suslin’s theorem, which states that if a set and its complement are both analytic, thenthe set is Borel.

129.1 Notes[1] (Kechris 1995, p. 87).

[2] (Lusin 1927).

129.2 References• Kechris, Alexander (1995), Classical descriptive set theory, Graduate texts inmathematics 156, Berlin–Heidelberg–New York: Springer–Verlag, pp. xviii+402, doi:10.1007/978-1-4612-4190-4, ISBN 0-387-94374-9, MR1321597, Zbl 0819.04002 (ISBN 3-540-94374-9 for the European edition)

• Lusin, Nicolas (1927), “Sur les ensembles analytiques” (PDF), Fundamenta Mathematicae (in French) 10:1–95, JFM 53.0171.05.

147

Chapter 130

Lévy hierarchy

In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy offormulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language ofset theory. This is analogous to the arithmetical hierarchy which provides the classifications but for sentences of thelanguage of arithmetic.

130.1 Definitions

In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and respectivelyset membership predicates.The first level of the Levy hierarchy is defined as containing only formulas with no unbounded quantifiers, and isdenoted by ∆0 = Σ0 = Π0 .[1] The next levels are given by finding an equivalent formula in Prenex normal form,and counting the number of changes of quantifiers:In the theory ZFC, a formula A is called:[1]

Σi+1 if A is equivalent to ∃x1...∃xnB in ZFC, where B is Πi

Πi+1 if A is equivalent to ∀x1...∀xnB in ZFC, where B is Σi

If a formula is both Σi and Πi , it is called ∆i . As a formula might have several different equivalent formulas inPrenex normal form, it might belong to several different levels of the hierarchy. In this case, the lowest possible levelis the level of the formula.The Lévy hierarchy is sometimes defined for other theories S. In this case Σi and Πi by themselves refer only toformulas that start with a sequence of quantifiers with at most i−1 alternations, and ΣS

i and ΠSi refer to formulas

equivalent to Σi and Πi formulas in the theory S. So strictly speaking the levels Σi and Πi of the Lévy hierarchy forZFC defined above should be denoted by ΣZFC

i and ΠZFCi .

130.2 Examples

130.2.1 Σ0=Π0=Δ0 formulas and concepts

• x = y, z

• x ⊆ y

• x is a transitive set

• x is an ordinal, x is a limit ordinal, x is a successor ordinal

• x is a finite ordinal

• The first countable ordinal ω.

148

130.2. EXAMPLES 149

• f is a function. The range and domain of a function. The value of a function on a set.

• The product of two sets.

• The union of a set.

130.2.2 Δ1-formulas and concepts

• x is a well-founded relation on y

• x is finite

• Ordinal addition and multiplication and exponentiation

• The rank of a set

• The transitive closure of a set

130.2.3 Σ1-formulas and concepts

• x is countable

• |X|≤|Y |, |X|=|Y |

• x is constructible

130.2.4 Π1-formulas and concepts

• x is a cardinal

• x is a regular cardinal

• x is a limit cardinal

• x is an inaccessible cardinal.

• x is the powerset of y

130.2.5 Δ2-formulas and concepts

• κ is γ-supercompact

130.2.6 Σ2-formulas and concepts

• the Continuum Hypothesis

• there exists an inaccessible cardinal

• there exists a measurable cardinal

• κ is an n-huge cardinal

130.2.7 Π2-formulas and concepts

• The axiom of constructibility: V = L

150 CHAPTER 130. LÉVY HIERARCHY

130.2.8 Δ3-formulas and concepts

130.2.9 Σ3-formulas and concepts

• There is a supercompact cardinal

130.2.10 Π3-formulas and concepts

• κ is an extendible cardinal

130.2.11 Σ4-formulas and concepts

• There is a extendible cardinal

130.3 Properties

Jech p. 184 Devlin p. 29

130.4 See also• arithmetic hierarchy

• Absoluteness

130.5 References[1] Walicki, Michal (2012). Mathematical Logic, p. 225. World Scientific Publishing Co. Pte. Ltd. ISBN 9789814343862

• Devlin, Keith J. (1984). Constructibility. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. pp.27–30. Zbl 0542.03029.

• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. p. 183. ISBN 978-3-540-44085-7. Zbl 1007.03002.

• Kanamori, Akihiro (2006). “Levy and set theory” (PDF). Annals of Pure and Applied Logic 140: 233–252.doi:10.1016/j.apal.2005.09.009. Zbl 1089.03004.

• Levy, Azriel (1965). A hierarchy of formulas in set theory. Mem. Am. Math. Soc. 57. Zbl 0202.30502.

Chapter 131

Martin’s maximum

In set theory, a branch of mathematical logic, Martin’s maximum, introduced by Foreman, Magidor & Shelah(1988), is a generalization of the proper forcing axiom, itself a generalization of Martin’s axiom. It represents thebroadest class of forcings for which a forcing axiom is consistent.Martin’s maximum (MM) states that if D is a collection of ℵ1 dense subsets of a notion of forcing that preservesstationary subsets of ω1, then there is a D-generic filter. It is a well known fact that forcing with a ccc notion offorcing preserves stationary subsets of ω1, thus MM extends MA( ℵ1 ). If (P,≤) is not a stationary set preservingnotion of forcing, i.e., there is a stationary subset of ω1, which becomes nonstationary when forcing with (P,≤), thenthere is a collection D of ℵ1 dense subsets of (P,≤), such that there is no D-generic filter. This is why MM is calledthe maximal extension of Martin’s axiom.The existence of a supercompact cardinal implies the consistency of Martin’s maximum.[1] The proof uses Shelah'stheories of semiproper forcing and iteration with revised countable supports.MM implies that the value of the continuum is ℵ2

[2] and that the ideal of nonstationary sets on ω1 is ℵ2 -saturated.[3]It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ≥ω2 and every elementof S has countable cofinality, then there is an ordinal α<κ such that S∩α is stationary in α. In fact, S contains a closedsubset of order type ω1.

131.1 References[1] Jech (2003) p.684

[2] Jech (2003) p.685

[3] Jech (2003) p.687

• Foreman, M.; Magidor, M.; Shelah, Saharon (1988), “Martin’s maximum, saturated ideals, and nonregular ul-trafilters. I.”,Ann. OfMath. (TheAnnals ofMathematics, Vol. 127, No. 1) 127 (1): 1–47, doi:10.2307/1971415,JSTOR 1971415, MR 0924672, Zbl 0645.03028 correction

• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third millennium ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7, Zbl 1007.03002

• Moore, Justin Tatch (2011), “Logic and foundations: the proper forcing axiom”, in Bhatia, Rajendra, Proceedingsof the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: In-vited lectures (PDF), Hackensack, NJ: World Scientific, pp. 3–29, ISBN 978-981-4324-30-4, Zbl 1258.03075

131.2 See also

Transfinite number

151

Chapter 132

Material nonimplication

Venn diagram of A ↛ B∧⇔⇔ ¬

Material nonimplication or abjunction (latin ab = “from”, junctio =–"joining”) is the negation of material impli-cation. That is to say that for any two propositions P and Q, the material nonimplication from P to Q is true if andonly if not P implies Q. This is more naturally stated as that the material nonimplication from P to Q is true is onlytrue if P is true and Q is false.It may be written using logical notation as:

p⊅qLpqp↛q

And is equivalent to:

152

132.1. DEFINITION 153

p∧~q

132.1 Definition

132.1.1 Truth table

132.2 Properties

falsehood-preserving: The interpretation under which all variables are assigned a truth value of “false” produces atruth value of “false” as a result of material nonimplication.

132.3 Symbol

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is8603 (decimal).

132.4 Natural language

132.4.1 Grammatical

132.4.2 Rhetorical

“p but not q.”

132.5 Boolean algebra

Further information: Boolean algebra

(A'+B)'

132.6 Computer science

Bitwise operation: A&(~B)Logical operation: A&&(!B)

132.7 See also• Implication

132.8 References

Chapter 133

Maximal set

In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerablesubset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers,either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within thelattice of the recursively enumerable sets.Maximal sets have many interesting properties: they are simple, hypersimple, hyperhypersimple and r-maximal; thelatter property says that every recursive set R contains either only finitely many elements of the complement of A oralmost all elements of the complement of A. There are r-maximal sets that are not maximal; some of them do evennot have maximal supersets. Myhill (1956) asked whether maximal sets exist and Friedberg (1958) constructed one.Soare (1974) showed that the maximal sets form an orbit with respect to automorphism of the recursively enumerablesets under inclusion (modulo finite sets). On the one hand, every automorphism maps a maximal set A to anothermaximal set B; on the other hand, for every two maximal sets A, B there is an automorphism of the recursivelyenumerable sets such that A is mapped to B.

133.1 References• Friedberg, Richard M. (1958), “Three theorems on recursive enumeration. I. Decomposition. II. Maximal set.III. Enumeration without duplication”, The Journal of Symbolic Logic (Association for Symbolic Logic) 23 (3):309–316, doi:10.2307/2964290, JSTOR 2964290, MR 0109125

• Myhill, John (1956), “Solution of a problem of Tarski”, The Journal of Symbolic Logic (Association for Sym-bolic Logic) 21 (1): 49–51, doi:10.2307/2268485, JSTOR 2268485, MR 0075894

• H. Rogers, Jr., 1967. The Theory of Recursive Functions and Effective Computability, second edition 1987,MIT Press. ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1.

• Soare, Robert I. (1974), “Automorphisms of the lattice of recursively enumerable sets. I.Maximal sets”,Annalsof Mathematics. Second Series (Annals of Mathematics) 100 (1): 80–120, doi:10.2307/1970842, JSTOR1970842, MR 0360235

154

Chapter 134

Michael D. Morley

Michael Morley in Berkeley

Michael DarwinMorley (born 1930) is an American mathematician, currently professor emeritus at Cornell Univer-sity. His research is in advanced mathematical logic and model theory, and he is best known for Morley’s categoricitytheorem, which he proved in his Ph.D. thesis “Categoricity in Power” in 1962.His formal Ph.D. advisor was Saunders MacLane at the University of Chicago, but he actually finished his thesisunder the guidance of Robert Vaught at the University of California, Berkeley.

134.1 See also

• Morley’s problem

155

156 CHAPTER 134. MICHAEL D. MORLEY

134.2 External links• Math genealogy database

• Morley’s home page

Chapter 135

Milner–Rado paradox

In set theory, a branch of mathematics, the Milner – Rado paradox, found by Eric Charles Milner and RichardRado (1965), states that every ordinal number α less than the successor κ+ of some cardinal number κ can be writtenas the union of sets X1,X2,... where Xn is of order type at most κn for n a positive integer.

135.1 Proof

The proof is by transfinite induction. Let α be a limit ordinal (the induction is trivial for successor ordinals), and foreach β < α , let Xβ

nn be a partition of β satisfying the requirements of the theorem.Fix an increasing sequence βγγ<cf (α) cofinal in α with β0 = 0 .Note cf (α) ≤ κ .Define:

Xα0 = 0; Xα

n+1 =∪γ

Xβγ+1n \ βγ

Observe that:

∪n>0

Xαn =

∪n

∪γ

Xβγ+1n \ βγ =

∪γ

∪n

Xβγ+1n \ βγ =

∪γ

βγ+1 \ βγ = α \ β0

and so∪

nXαn = α .

Let ot (A) be the order type of A . As for the order types, clearly ot(Xα0 ) = 1 = κ0 .

Noting that the sets βγ+1 \ βγ form a consecutive sequence of ordinal intervals, and that each Xβγ+1n \ βγ is a tail

segment of Xβγ+1n we get that:

ot(Xαn+1) =

∑γ

ot(Xβγ+1n \ βγ) ≤

∑γ

κn = κn · cf(α) ≤ κn · κ = κn+1

135.2 References• Milner, E. C.; Rado, R. (1965), “The pigeon-hole principle for ordinal numbers”, Proc. London Math. Soc.

(3) 15: 750–768, doi:10.1112/plms/s3-15.1.750, MR 0190003

• How to prove Milner-Rado Paradox? - Mathematics Stack Exchange

157

Chapter 136

Minimal logic

Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It isa variant of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does),but also the principle of explosion (ex falso quodlibet).Just like intuitionistic logic, minimal logic can be formulated in a language using→, ∧, ∨, ⊥ (implication, conjunction,disjunction and falsum) as the basic connectives, treating ¬A as an abbreviation for (A → ⊥). In this language it isaxiomatized by the positive fragment (i.e., formulas using only →, ∧, ∨) of intuitionistic logic, with no additionalaxioms or rules about ⊥. Thus minimal logic is a subsystem of intuitionistic logic, and it is strictly weaker as it doesnot derive the ex falso quodlibet principle ¬A,A ⊢ B (however, it derives its special case ¬A,A ⊢ ¬B ).Adding the ex falso axiom ¬A → (A → B) to minimal logic results in intuitionistic logic, and adding the doublenegation law ¬¬A→ A to minimal logic results in classical logic.Minimal logic is closely related to simply typed lambda calculus via the Curry-Howard isomorphism, i.e. the typingderivations of simply typed lambda terms are isomorphic to natural deduction proofs in minimal logic.

136.1 References• Johansson, Ingebrigt, 1936, "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus.” Compositio

Mathematica 4, 119–136.

158

Chapter 137

Nice name

In set theory, a nice name is used in forcing to impose an upper bound on the number of subsets in the generic model.It is used in the context of forcing to prove independence results in set theory such as Easton’s theorem.

137.1 Formal definition

LetM |= ZFC be transitive, (P, <) a forcing notion inM , and suppose G ⊆ P is generic overM . Then for any P-name inM , τ ,η is a nice name for a subset of τ if η is a P -name satisfying the following properties:(1) dom(η) ⊆ dom(τ)

(2) For all P -names σ ∈M , p ∈ P|⟨σ, p⟩ ∈ η forms an antichain.(3) (Natural addition): If ⟨σ, p⟩ ∈ η , then there exists q ≥ p in P such that ⟨σ, q⟩ ∈ τ .

137.2 References• Kenneth Kunen (1980) Set theory: an introduction to independence proofs, Volume 102 of Studies in logic andthe foundations of mathematics (Elsevier) ISBN 0-444-85401-0, p.208

159

Chapter 138

Normal measure

In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of theidentity function on κ maps to κ itself in the ultrapower construction. Equivalently, if f:κ→κ is such that f(α)<α formost α<κ, then there is a β<κ such that f(α)=β for most α<κ. (Here, “most” means that the set of elements of κwhere the property holds is a member of the ultrafilter, i.e. has measure 1.) Also equivalent, the ultrafilter (set ofsets of measure 1) is closed under diagonal intersection.For a normal measure, any closed unbounded (club) subset of κ contains most ordinals less than κ. And any subsetcontaining most ordinals less than κ is stationary in κ.If an uncountable cardinal κ has a measure on it, then it has a normal measure on it.

138.1 See also• Measurable cardinal

• Club set

138.2 References• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (1sted.). Springer. ISBN 3-540-57071-3. pp 52–53

160

Chapter 139

Omega-categorical theory

Inmathematical logic, an omega-categorical theory is a theory that has only one countablemodel up to isomorphism.Omega-categoricity is the special case κ = ℵ0 = ω of κ-categoricity, and omega-categorical theories are also referredto as ω-categorical. The notion is most important for countable first-order theories.

139.1 Equivalent conditions for omega-categoricity

Many conditions on a theory are equivalent to the property of omega-categoricity. In 1959 Erwin Engeler, CzesławRyll-Nardzewski and Lars Svenonius, proved several independently.[1] Despite this, the literature still widely refers tothe Ryll-Nardzewski theorem as a name for these conditions. The conditions included with the theorem vary betweenauthors.[2][3]

Given a countable complete first-order theory T with infinite models, the following are equivalent:

• The theory T is omega-categorical.

• Every countable model of T has an oligomorphic automorphism group.

• Some countable model of T has an oligomorphic automorphism group.[4]

• The theory T has a model which, for every natural number n, realizes only finitely many n-types, that is, theStone space Sn(T) is finite.

• For every natural number n, T has only finitely many n-types.

• For every natural number n, every n-type is isolated.

• For every natural number n, up to equivalence modulo T there are only finitely many formulas with n freevariables, in other words, every nth Lindenbaum-Tarski algebra of T is finite.

• Every model of T is atomic.

• Every countable model of T is atomic.

• The theory T has a countable atomic and saturated model.

• The theory T has a saturated prime model.

139.2 Notes[1] Rami Grossberg, José Iovino and Olivier Lessmann, A primer of simple theories

[2] Hodges, Model Theory, p. 341.

[3] Rothmaler, p. 200.

[4] Cameron (1990) p.30

161

162 CHAPTER 139. OMEGA-CATEGORICAL THEORY

139.3 References• Cameron, Peter J. (1990), Oligomorphic permutation groups, London Mathematical Society Lecture Note Se-ries 152, Cambridge: Cambridge University Press, ISBN 0-521-38836-8, Zbl 0813.20002

• Chang, Chen Chung; Keisler, H. Jerome (1989) [1973], Model Theory, Elsevier, ISBN 978-0-7204-0692-4

• Hodges, Wilfrid (1993), Model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-30442-9

• Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6

• Poizat, Bruno (2000), A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Berlin,New York: Springer-Verlag, ISBN 978-0-387-98655-5

• Rothmaler, Philipp (2000), Introduction to Model Theory, New York: Taylor & Francis Group, ISBN 978-90-5699-313-9

Chapter 140

Ordinal definable set

In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finitenumber of ordinals by a first order formula. Ordinal definable sets were introduced by Gödel (1965).A drawback to this informal definition is that requires quantification over all first order formulas, which cannot beformalized in the language of set theory. However there is a different way of stating the definition that can be soformalized. In this approach, a set S is formally defined to be ordinal definable if there is some collection of ordinalsα1...α such that S ∈ Vα1 and S can be defined as an element of Vα1 by a first-order formula φ taking α2...α asparameters. Here Vα1 denotes the set indexed by the ordinal α1 in the von Neumann hierarchy of sets. In otherwords, S is the unique object such that φ(S, α2...α ) holds with its quantifiers ranging over Vα1 .The class of all ordinal definable sets is denoted OD; it is not necessarily transitive, and need not be a model ofZFC because it might not satisfy the axiom of extensionality. A set is hereditarily ordinal definable if it is ordinaldefinable and all elements of its transitive closure are ordinal definable. The class of hereditarily ordinal definable setsis denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axiomsof set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situationholds is referred to as V = OD or V = HOD. It follows from V = L, and is equivalent to the existence of a (definable)well-ordering of the universe. Note however that the formula expressing V = HOD need not hold true within HOD,as it is not absolute for models of set theory: within HOD, the interpretation of the formula for HOD may yield aneven smaller inner model.HOD has been found to be useful in that it is an inner model that can accommodate essentially all known largecardinals. This is in contrast with the situation for core models, as core models have not yet been constructed thatcan accommodate supercompact cardinals, for example.

140.1 References• Gödel, Kurt (1965) [1946], “Remarks before the Princeton Bicentennial Conference on Problems in Mathe-matics”, in Davis, Martin, The undecidable. Basic papers on undecidable propositions, unsolvable problems andcomputable functions, Raven Press, Hewlett, N.Y., pp. 84–88, ISBN 978-0-486-43228-1, MR 0189996

• Kunen, Kenneth (1980), Set theory: An introduction to independence proofs, Elsevier, ISBN 978-0-444-86839-8

163

Chapter 141

Ordinal logic

In mathematics, ordinal logic is a logic associated with an ordinal number by recursively adding elements to asequence of previous logics.[1][2] The concept was introduced in 1938 by Alan Turing in his PhD dissertation atPrinceton in view of Gödel’s incompleteness theorems.[3][1]

While Gödel showed that every system of logic suffers from some form of incompleteness, Turing focused on amethod so that from a given system of logic a more complete system may be constructed. By repeating the processa sequence L1, L2, … of logics is obtained, each more complete than the previous one. A logic L can then beconstructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc.Thus Turing showed how one can associate a logic with any constructive ordinal.[3]

141.1 References[1] Solomon Feferman, Turing in the Land of O(z) in “The universal Turing machine: a half-century survey” by Rolf Herken

1995 ISBN 3-211-82637-8 page 111

[2] Concise Routledge encyclopedia of philosophy 2000 ISBN 0-415-22364-4 page 647

[3] Alan Turing, Systems of Logic Based on Ordinals Proceedings London Mathematical Society Volumes 2–45, Issue 1, pp.161–228.

164

Chapter 142

Paraconsistent mathematics

Paraconsistent mathematics (sometimes called inconsistent mathematics) represents an attempt to develop theclassical infrastructure of mathematics (e.g. analysis) based on a foundation of paraconsistent logic instead of classicallogic. A number of reformulations of analysis can be developed, for example functions which both do and do nothave a given value simultaneously.Chris Mortensen claims (see references):

One could hardly ignore the examples of analysis and its special case, the calculus. There prove to bemany places where there are distinctive inconsistent insights; see Mortensen (1995) for example. (1)Robinson’s non-standard analysis was based on infinitesimals, quantities smaller than any real number,as well as their reciprocals, the infinite numbers. This has an inconsistent version, which has some advan-tages for calculation in being able to discard higher-order infinitesimals. The theory of differentiationturned out to have these advantages, while the theory of integration did not. (2)

142.1 References• Inconsistent Mathematics, by Chris Mortensen, Dordrecht, Kluwer Academic Publishers, 1995 Kluwer Math-

ematics and Its Applications Series, Vol 312 ISBN 0-7923-3186-9

142.2 External links• Entry in the Internet Encyclopedia of Philosophy

• Entry in the Stanford Encyclopedia of Philosophy

• Lectures by Manuel Bremer of the University of Düsseldorf

165

Chapter 143

Polyadic algebra

Polyadic algebras (more recently called Halmos algebras[1]) are algebraic structures introduced by Paul Halmos.They are related to first-order logic in a way analogous to the relationship between Boolean algebras and propositionallogic (see Lindenbaum-Tarski algebra).There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras[1] (when equality ispart of the logic) and Lawvere's functorial semantics (categorical approach).[2]

143.1 References[1] Michiel Hazewinkel (2000). Handbook of algebra 2. Elsevier. pp. 87–89. ISBN 978-0-444-50396-1.

[2] Jon Barwise (1989). Handbook of mathematical logic. Elsevier. p. 293. ISBN 978-0-444-86388-1.

143.2 Further reading• Paul Halmos, Algebraic Logic, Chelsea Publishing, New York (1962)

166

Chapter 144

Predicate logic

For the specific term, see First-order logic.

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic, or infinitary logic. This formal system is distinguished from other systems in that itsformulae contain variables which can be quantified. Two common quantifiers are the existential ∃ (“there exists”)and universal ∀ (“for all”) quantifiers. The variables could be elements in the universe under discussion, or perhapsrelations or functions over that universe. For instance, an existential quantifier over a function symbol would beinterpreted as modifier “there is a function”. The foundations of predicate logic were developed independently byGottlob Frege and Charles Sanders Peirce.[1]

In informal usage, the term “predicate logic” occasionally refers to first-order logic. Some authors consider thepredicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from aninformal, more intuitive development.[2]

Predicate logics also include logics mixing modal operators and quantifiers. See Modal logic, Saul Kripke, BarcanMarcus formulae, A. N. Prior, and Nicholas Rescher.

144.1 See also

• First-order logic

• Propositional logic

• Existential graph

144.2 Footnotes[1] Eric M. Hammer: Semantics for Existential Graphs, Journal of Philosophical Logic, Volume 27, Issue 5 (October 1998),

page 489: “Development of first-order logic independently of Frege, anticipating prenex and Skolem normal forms”

[2] Among these authors is Stolyar, p. 166. Hamilton considers both to be calculi but divides them into an informal calculusand a formal calculus.

144.3 References

• A. G. Hamilton 1978, Logic for Mathematicians, Cambridge University Press, Cambridge UK ISBN 0-521-21838-1

• Abram Aronovic Stolyar 1970, Introduction to Elementary Mathematical Logic, Dover Publications, Inc. NY.ISBN 0-486-645614

167

168 CHAPTER 144. PREDICATE LOGIC

• George F Luger, Artificial Intelligence, Pearson Education, ISBN 978-81-317-2327-2

• Hazewinkel, Michiel, ed. (2001), “Predicate calculus”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 145

Principle of distributivity

The principle of distributivity states that the algebraic distributive law is valid for classical logic, where both logicalconjunction and logical disjunction are distributive over each other so that for any propositions A, B and C theequivalences

A ∧ (B ∨ C) ⇐⇒ (A ∧B) ∨ (A ∧ C)

and

A ∨ (B ∧ C) ⇐⇒ (A ∨B) ∧ (A ∨ C)

hold.The principle of distributivity is valid in classical logic, but invalid in quantum logic. The article Is logic empirical?discusses the case that quantum logic is the correct, empirical logic, on the grounds that the principle of distributivityis inconsistent with a reasonable interpretation of quantum phenomena.

169

Chapter 146

Projection (set theory)

In set theory, a projection is one of two closely related types of functions or operations, namely:

• A set-theoretic operation typified by the jth projectionmap, written projj , that takes an element x = (x1, . . . , xj , . . . , xk)

of the Cartesian product (X1 × · · · ×Xj × · · · ×Xk) to the value projj(x) = xj .[1]

• A function that sends an element x to its equivalence class under a specified equivalence relation E,[2] or,equivalently, a surjection from a set to another set.[3] The function from elements to equivalence classes is asurjection, and every surjection corresponds to an equivalence relation under which two elements are equivalentwhen they have the same image. The result of the mapping is written as [x] when E is understood, or writtenas [x]E when it is necessary to make E explicit.

146.1 See also• Cartesian product

• Projection (relational algebra)

• Projection (mathematics)

• Relation

146.2 References[1] Halmos, P. R. (1960), Naive Set Theory, Undergraduate Texts in Mathematics, Springer, p. 32, ISBN 9780387900926.

[2] Brown, Arlen; Pearcy, Carl M. (1995), An Introduction to Analysis, Graduate Texts in Mathematics 154, Springer, p. 8,ISBN 9780387943695.

[3] Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Springer Monographs in Mathematics, Springer, p. 34,ISBN 9783540440857.

170

Chapter 147

Proof compression

In proof theory, an area of mathematical logic, proof compression is the problem of algorithmically compressingformal proofs. The developed algorithms can be used to improve the proofs generated by automated theorem provingtools such as sat-solvers, SMT-solvers, first-order theorem provers and proof assistants.

147.1 Problem Representation

In propositional logic a resolution proof of a clause κ from a set of clauses C is a directed acyclic graph (DAG): theinput nodes are axiom inferences (without premises) whose conclusions are elements of C, the resolvent nodes areresolution inferences, and the proof has a node with conclusion κ .[1]

The DAG contains an edge from a node η1 to a node η2 if and only if a premise of η1 is the conclusion of η2 . Inthis case, η1 is a child of η2 , and η2 is a parent of η1 . A node with no children is a root.A proof compression algorithm will try to create a new DAG with fewer nodes that represents a valid proof of κ or,in some cases, a valid proof of a subset of κ .

147.1.1 A simple example

Let’s take a resolution proof for the clause a, b, c from the set of clauses

η1 : a, b, p , η2 : c,¬p η1 : a, b, p η2 : c,¬pη3 : a, b, c

p

Here we can see:

• η1 and η2 are input nodes.

• The node η3 has a pivot p ,

• left resolved literal p• right resolved literal ¬p

• η3 conclusion is the clause a, b, c

• η3 premises are the conclusion of nodes η1 and η2 (its parents)

• The DAG would be

η1 η2η3

171

172 CHAPTER 147. PROOF COMPRESSION

• η1 and η2 are parents of η3• η3 is a child of η1 and η2• η3 is a root of the proof

A (resolution) refutation of C is a resolution proof of ⊥ from C. It is a common that given a node η , to refer to theclause η or η ’s clause meaning the conclusion clause of η , and (sub)proof η meaning the (sub)proof having η as itsonly root.In some works it can be found an algebraic representation of a resolution inference. The resolvent of κ1 and κ2 withpivot p can be denoted as κ1 ⊙p κ2 . When the pivot is uniquely defined or irrelevant, we omit it and write simplyκ1 ⊙ κ2 . In this way, the set of clauses can be seen as an algebra with a commutative operator; and terms in thecorresponding term algebra denote resolution proofs in a notation style that is more compact and more convenientfor describing resolution proofs than the usual graph notation.In our last example the notation of the DAG would be a, b, p ⊙p c,¬p or simply a, b, p ⊙ c,¬p .

We can identifyη1︷ ︸︸ ︷

a, b, p⊙

η2︷ ︸︸ ︷c,¬p︸ ︷︷ ︸

η3

147.2 Compression algorithms

Algorithms for compression of sequent calculus proofs include Cut-introduction and Cut-elimination.Algorithms for compression of propositional resolution proofs includeRecycleUnits,[2] RecyclePivots,[3] RecyclePivotsWithIntersection,[4]LowerUnits,[5] LowerUnivalents,[6] Split,[7] Reduce&Reconstruct,[8] and Subsumption.

147.3 Notes[1] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial

Regularization. 23rd International Conference on Automated Deduction, 2011.

[2] Bar-Ilan, O.; Fuhrmann, O.; Hoory, S. ; Shacham, O. ; Strichman, O. Linear-time Reductions of Resolution Proofs. Hard-ware and Software: Verification and Testing, p. 114–128, Springer, 2011.

[3] Bar-Ilan, O.; Fuhrmann, O.; Hoory, S. ; Shacham, O. ; Strichman, O. Linear-time Reductions of Resolution Proofs. Hard-ware and Software: Verification and Testing, p. 114–128, Springer, 2011.

[4] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via PartialRegularization. 23rd International Conference on Automated Deduction, 2011.

[5] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via PartialRegularization. 23rd International Conference on Automated Deduction, 2011.

[6] https://github.com/Paradoxika/Skeptik/tree/develop/doc/papers/LUniv

[7] Cotton, Scott. “Two Techniques for Minimizing Resolution Proofs”. 13th International Conference on Theory and Appli-cations of Satisfiability Testing, 2010.

[8] Simone, S.F. ; Brutomesso, R. ; Sharygina, N. “An Efficient and Flexible Approach to Resolution Proof Reduction”. 6thHaifa Verification Conference, 2010.

Chapter 148

Proof mining

In proof theory, a branch of mathematical logic, proof mining (or unwinding) is a research program that analyzesformalized proofs, especially in analysis, to obtain explicit bounds or rates of convergence from proofs that, whenexpressed in natural language, appear to be nonconstructive.[1] This research has led to improved results in analysisobtained from the analysis of classical proofs.

148.1 References[1] Ulrich Kohlenbach (2008). Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Verlag,

Berlin. pp. 1–536.

• Ulrich Kohlenbach and Paulo Oliva, “Proof Mining: A systematic way of analysing proofs in mathematics”,Proc. Steklov Inst. Math, 242:136–164, 2003

• Paulo Oliva, “Proof Mining in Subsystems of Analysis”, BRICS PhD thesis citeseer

173

Chapter 149

Pseudo-intersection

In mathematical set theory, a pseudo-intersection of a family of sets is an infinite set S such that each element ofthe family contains all but a finite number of elements of S. The pseudo-intersection number, sometimes denotedby the fraktur letter 𝔭, is the smallest size of a family of infinite subsets of the natural numbers that has the strongfinite intersection property but has no pseudo-intersection.

149.1 References• Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001

174

Chapter 150

Pseudo-order

In constructive mathematics, a pseudo-order is a constructive generalisation of a linear order to the continuouscase. The usual trichotomy law does not hold in the constructive continuum because of its indecomposability, so thiscondition is weakened.A pseudo-order is a binary relation satisfying the following conditions:

1. It is not possible for two elements to each be less than the other. That is, ∀x, y : ¬ (x < y ∧ y < x) .2. For all x, y, and z, if x < y then either x < z or z < y. That is, ∀x, y, z : x < y → (x < z ∨ z < y) .3. Every two elements for which neither one is less than the other must be equal. That is, ∀x, y : ¬ (x < y ∨ y <x) → x = y

This first condition is simply antisymmetry. It follows from the first two conditions that a pseudo-order is transitive.The second condition is often called co-transitivity or comparison and is the constructive substitute for trichotomy. Ingeneral, given two elements of a pseudo-ordered set, it is not always the case that either one is less than the other orelse they are equal, but given any nontrivial interval, any element is either above the lower bound, or below the upperbound.The third condition is often taken as the definition of equality. The natural apartness relation on a pseudo-ordered setis given by

x#y ↔ x < y ∨ y < x

and equality is defined by the negation of apartness.The negation of the pseudo-order is a partial order which is close to a total order: if x ≤ y is defined as the negationof y < x, then we have

¬ (¬ (x ≤ y) ∧ ¬ (y ≤ x)).

Using classical logic one would then conclude that x ≤ y or y ≤ x, so it would be a total order. However, this inferenceis not valid in the constructive case.The prototypical pseudo-order is that of the real numbers: one real number is less than another if there exists (onecan construct) a rational number greater than the former and less than the latter. In other words, x < y if there existsa rational number z such that x < z < y.

150.1 References• Arend Heyting (1966) Intuitionism: An introduction. Second revised edition North-Holland Publishing Co.,Amsterdam.

http://books.google.com/books/about/Intuitionism.html?id=4rhLAAAAMAAJ

175

Chapter 151

Ramified forcing

In the mathematical discipline of set theory, ramified forcing is the original form of forcing introduced by Cohen(1963) to prove the independence of the continuum hypothesis from Zermelo–Fraenkel set theory. Ramified forcingstarts with a model M of set theory in which the axiom of constructibility, V = L, holds, and then builds up a largermodel M[G] of Zermelo–Fraenkel set theory by adding a generic subset G of a partially ordered set to M, imitatingKurt Gödel's constructible hierarchy.Dana Scott and Robert Solovay realized that the use of constructible sets was an unnecessary complication, and couldbe replaced by a simpler construction similar to John von Neumann's construction of the universe as a union of setsR(α) for ordinals α. Their simplification was originally called “unramified forcing” (Shoenfield 1971), but is nowusually just called “forcing”. As a result, ramified forcing is only rarely used.

151.1 References• Cohen, P. J. (1966), Set Theory and the Continuum Hypothesis, Menlo Park, CA: W. A. Benjamin.

• Cohen, Paul J. (1963), “The Independence of the ContinuumHypothesis”, Proceedings of the National Academyof Sciences of the United States of America 50 (6): 1143–1148, doi:10.1073/pnas.50.6.1143, ISSN 0027-8424,JSTOR 71858, PMC 221287, PMID 16578557.

• Shoenfield, J. R. (1971), “Unramified forcing”, Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII, PartI, Providence, R.I.: Amer. Math. Soc., pp. 357–381, MR 0280359.

176

Chapter 152

Ramsey cardinal

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962)and named after Frank P. Ramsey.With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number κ such that for every function

f: [κ]<ω → 0, 1

there is a set A of cardinality κ that is homogeneous for f (i.e.: for every n, f is constant on the subsets of cardinalityn from A) is called Ramsey. A cardinal κ is called almost Ramsey if for every function

f: [κ]<ω → 0, 1

and for every λ < κ, there is a set of order type λ that is homogeneous for f.The existence of a Ramsey cardinal is sufficient to prove the existence of 0#. In fact, if κ is Ramsey, then every setwith rank less than κ has a sharp.Every measurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal.A property intermediate in strength between Ramseyness and measurability is existence of a κ-complete normalnon-principal ideal I on κ such that for every A ∉ I and for every function

f: [κ]<ω → 0, 1

there is a set B ⊂ A not in I that is homogeneous for f. If I is taken to be the ideal of nonstationary sets, this propertydefines the ineffably Ramsey cardinals.The existence of Ramsey cardinal implies that the existence of the zero sharp cardinal and this in turn implies thefalsity of Axiom of Constructibility of Kurt Gödel.

152.1 References• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of

Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

• Erdős, Paul; Hajnal, András (1962), “Some remarks concerning our paper “On the structure of set-mappings.Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal”, Acta MathematicaAcademiae Scientiarum Hungaricae 13: 223–226, doi:10.1007/BF02033641, ISSN 0001-5954, MR 0141603

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

177

Chapter 153

Rank-into-rank

In set theory, a branch of mathematics, a rank-into-rank is a large cardinal λ satisfying one of the following fouraxioms given in order of increasing consistency strength. (They are sometimes known as rank-into-rank embeddings,where a rank is one of the sets Vλ of the von Neumann hierarchy.)

• Axiom I3: There is a nontrivial elementary embedding of Vλ into itself.

• Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ whereλ is the first fixed point above the critical point.

• Axiom I1: There is a nontrivial elementary embedding of Vλ₊₁ into itself.

• Axiom I0: There is a nontrivial elementary embedding of L(Vλ₊₁) into itself with the critical point below λ.

These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom forReinhardt cardinals is stronger, but is not consistent with the axiom of choice.If j is the elementary embedding mentioned in one of these axioms and κ is its critical point, then λ is the limit ofjn(κ) as n goes to ω. More generally, if the axiom of choice holds, it is provable that if there is a nontrivial elementaryembedding of Vα into itself then α is either a limit ordinal of cofinality ω or the successor of such an ordinal.The axioms I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen’sinconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them,but this has not yet happened and they are now usually believed to be consistent.Every I0 cardinal κ (speaking here of the critical point of j) is an I1 cardinal.Every I1 cardinal κ is an I2 cardinal and has a stationary set of I2 cardinals below it.Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it.Every I3 cardinal κ has another I3 cardinal above it and is an n-huge cardinal for every n<ω.Axiom I1 implies that Vλ₊₁ (equivalently, H(λ+)) does not satisfy V=HOD. There is no set S⊂λ definable in Vλ₊₁ (evenfrom parameters Vλ and ordinals <λ+) with S cofinal in λ and |S|<λ, that is, no such S witnesses that λ is singular.And similarly for Axiom I0 and ordinal definability in L(Vλ₊₁) (even from parameters in Vλ). However globally, andeven in Vλ,[1] V=HOD is relatively consistent with Axiom I1.

153.1 References

• Gaifman, Haim (1974), “Elementary embeddings of models of set-theory and certain subtheories”, Axiomaticset theory, Proc. Sympos. Pure Math., XIII, Part II, Providence R.I.: Amer. Math. Soc., pp. 33–101, MR0376347

• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.), Springer, ISBN 3-540-00384-3

178

153.1. REFERENCES 179

• Laver, Richard (1997), “Implications between strong large cardinal axioms”, Ann. Pure Appl. Logic 90 (1-3):79–90, doi:10.1016/S0168-0072(97)00031-6, MR 1489305

• Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978), “Strong axioms of infinity and ele-mentary embeddings”, Annals of Mathematical Logic 13 (1): 73–116, doi:10.1016/0003-4843(78)90031-1

[1] Consistency of V = HOD With the Wholeness Axiom, Paul Corazza, Archive for Mathematical Logic, No. 39, 2000.

Chapter 154

Recursive ordinal

In mathematics, specifically set theory, an ordinal α is said to be recursive if there is a recursive well-ordering of asubset of the natural numbers having the order type α .It is trivial to check that ω is recursive, the successor of a recursive ordinal is recursive, and the set of all recursiveordinals is closed downwards. The supremumof all recursive ordinals is called the Church-Kleene ordinal and denotedby ωCK

1 . Indeed, an ordinal is recursive if and only if it is smaller than ωCK1 . Since there are only countably many

recursive relations, there are also only countably many recursive ordinals. Thus, ωCK1 is countable.

The recursive ordinals are exactly the ordinals that have an ordinal notation in Kleene’s O .

154.1 See also• Arithmetical hierarchy

• Large countable ordinals

• Ordinal notation

154.2 References• Rogers, H. The Theory of Recursive Functions and Effective Computability, 1967. Reprinted 1987, MIT Press,ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1

• Sacks, G. Higher Recursion Theory. Perspectives in mathematical logic, Springer-Verlag, 1990. ISBN 0-387-19305-7

180

Chapter 155

Reduced product

For the reduced product in algebraic topology, see James reduced product.

Inmodel theory, a branch ofmathematical logic, and in algebra, the reduced product is a construction that generalizesboth direct product and ultraproduct.Let Si | i ∈ I be a family of structures of the same signature σ indexed by a set I, and let U be a filter on I. Thedomain of the reduced product is the quotient of the Cartesian product

∏i∈I

Si

by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if

i ∈ I : ai = bi ∈ U

If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the originalCartesian product. If U is an ultrafilter, the reduced product is an ultraproduct.Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are inter-preted by

R((a1i )/∼, . . . , (ani )/∼) ⇐⇒ i ∈ I | RSi(a1i , . . . , ani ) ∈ U.

For example, if each structure is a vector space, then the reduced product is a vector space with addition defined as(a + b)i = ai + bi and multiplication by a scalar c as (ca)i = c ai.

155.1 References• Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]. Model Theory. Studies in Logic and the Foundationsof Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3., Chapter 6.

181

Chapter 156

Redundant proof

In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. Thatis, a proof ψ of κ is considered redundant if there exists another proof ψ′ of κ′ such that κ′ ⊆ κ (i.e. κ′ subsumes κ) and |ψ′| < |ψ| where |φ| is the number of nodes in φ .[1]

156.1 Local redundancy

A proof containing a subproof of the shapes (here omitted pivots indicate that the resolvents must be uniquely defined)

(η ⊙ η1)⊙ (η ⊙ η2) or η ⊙ (η1 ⊙ (η ⊙ η2))

is locally redundant.Indeed, both of these subproofs can be equivalently replaced by the shorter subproof η ⊙ (η1 ⊙ η2) . In the caseof local redundancy, the pairs of redundant inferences having the same pivot occur close to each other in the proof.However, redundant inferences can also occur far apart in the proof.The following definition generalizes local redundancy by considering inferences with the same pivot that occur withindifferent contexts. We write ψ [η] to denote a proof-context ψ [−] with a single placeholder replaced by the subproofη .

156.2 Global redundancy

A proof

ψ[ψ1[η ⊙p η1]⊙ ψ2[η ⊙p η2]] or ψ[ψ1[η ⊙p (η1 ⊙ ψ2[η ⊙p η2])]]

is potentially (globally) redundant. Furthermore, it is (globally) redundant if it can be rewritten to one of the followingshorter proofs:

ψ[η ⊙p (ψ1[η1]⊙ ψ2[η2])] or η ⊙p ψ[ψ1[η1]⊙ ψ2[η2]] or ψ[ψ1[η1]⊙ ψ2[η2]].

156.2.1 Example

The proof

182

156.3. NOTES 183

η : p, q η1 : ¬p, rq, r

pη3 : ¬q

rq

η η2 : ¬p, s,¬rq, s,¬r

pη3

s,¬rq

ψ : sr

is locally redundant as it is an instance of the first pattern in the definition ((η ⊙p η1)⊙ η3)⊙ ((η ⊙p η2)⊙ η3).

• The pattern is ψ[ψ1[η ⊙p η1]⊙ ψ2[η ⊙p η2]]

• ψ1[−] = ψ2[−] = _⊙ η3 and ψ[−] = _

But it is not globally redundant because the replacement terms according to the definition contain ψ1[η1]⊙ψ2[η2] inall the cases and ψ1[η1]⊙ ψ2[η2] = (η1 ⊙ η3)⊙ (η2 ⊙ η3) does not correspond to a proof. In particular, neither η1nor η2 can be resolved with η3 , as they do not contain the literal q .The second pattern of potentially globally redundant proofs appearing in global redundancy definition is related to thewell-known notion of regularity. [This link to “regularity” is (obviously) a link to a disambiguation page.] Informally,a proof is irregular if there is a path from a node to the root of the proof such that a literal is used more than once asa pivot in this path.

156.3 Notes[1] Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. Compression of Propositional Resolution Proofs via Partial

Regularization. 23rd International Conference on Automated Deduction, 2011.

Chapter 157

Reflecting cardinal

In set theory, a mathematical discipline, a reflecting cardinal is a cardinal number κ for which there is a normalideal I on κ such that for every X∈I+, the set of α∈κ for which X reflects at α is in I+. (A stationary subset S of κ issaid to reflect at α<κ if S∩α is stationary in α.) Reflecting cardinals were introduced by (Mekler & Shelah 1989).Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. Every reflectingcardinal is a greatly Mahlo cardinal, and is also a limit of greatly Mahlo cardinals, where a cardinal κ is called greatlyMahlo if it is κ+-Mahlo.

157.1 References• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, NewYork: Springer-Verlag, p. 697, ISBN 978-3-540-44085-7

• Mekler, Alan H.; Shelah, Saharon (1989), “The consistency strength of ``every stationary set reflects", IsraelJournal of Mathematics 67 (3): 353–366, doi:10.1007/BF02764953, ISSN 0021-2172, MR 1029909

184

Chapter 158

Remarkable cardinal

In mathematics, a remarkable cardinal is a certain kind of large cardinal number.A cardinal κ is called remarkable if for all regular cardinals θ > κ, there exist π, M, λ, σ, N and ρ such that

1. π : M → Hθ is an elementary embedding

2. M is countable and transitive

3. π(λ) = κ

4. σ : M → N is an elementary embedding with critical point λ

5. N is countable and transitive

6. ρ = M ∩ Ord is a regular cardinal in N

7. σ(λ) > ρ

8. M = HᵨN , i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"

158.1 See also• Hereditarily countable set

158.2 References• Schindler, Ralf (2000), “Proper forcing and remarkable cardinals”, The Bulletin of Symbolic Logic 6 (2): 176–184, doi:10.2307/421205, ISSN 1079-8986, MR 1765054

185

Chapter 159

Richardson’s theorem

In mathematics, Richardson’s theorem establishes a limit on the extent to which an algorithm can decide whethercertainmathematical expressions are equal. It states that for a certain fairly natural class of expressions, it is undecidablewhether a particular expression E satisfies the equation E = 0, and similarly undecidable whether the functions de-fined by expressions E and F are everywhere equal (in fact E = F if and only if E - F = 0). It was proved in 1968 bycomputer scientist Daniel Richardson of the University of Bath.Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the numberπ, the number log 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin,exp, and abs functions.For some classes of expressions (generated by other primitives than in Richardson’s theorem) there exist algorithmsthat can determine whether an expression is zero.[1]

159.1 Statement of the theorem

Richardson’s theorem can be stated as follows:[2] Let E be a set of expressions in the variable x which contains x and,as constant expressions, all rational numbers, and is such that if A(x) and B(x) are in E, then A(x) + B(x), A(x) - B(x),A(x)B(x), and A(B(x)) are also in E. Then:

• if x, log 2, π, ex, sin x ∈ E, then the problem of determining, for an expression A(x) in E, whether A(x) < 0 forsome x is unsolvable;

• if also |x| ∈ E then the problem of determining whether A(x) = 0 for all x is also unsolvable;

• if furthermore there is a function B(x) ∈ E without an antiderivative in E then the integration problem isunsolvable. (Example: eax2 has an antiderivative in the elementary functions if and only if a = 0.)

159.2 Extensions

After Hilbert’s Tenth Problem was solved in 1970, B. F. Caviness observed that the use of ex and log 2 could beremoved.[3] P. S. Wang[4] later noted that under the same assumptions under which the question of whether there wasx with A(x) < 0 was insoluble, the question of whether there was x with A(x) = 0 was also insoluble.Miklós Laczkovich[5] removed also the need for π and reduced the use of composition. In particular, given anexpression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn), both the question of whether A(x) >0 for some x and whether A(x) = 0 for some x are unsolvable.By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is notpossible to remove the sine function entirely.

186

159.3. SEE ALSO 187

159.3 See also• Constant problem

159.4 References[1] The identity problem for elementary functions and constants by Richardson and Fitch (pdf file)

[2] “Some Undecidable Problems Involving Elementary Functions of a Real Variable”, Daniel Richardson, J. Symbolic Logic33, #4 (1968), pp. 514-520, JSTOR 2271358.

[3] On Canonical Forms and Simplification, B. F. Caviness, JACM, 17, #2 (April 1970), pp. 385-396.

[4] P. S. Wang, The undecidability of the existence of zeros of real elementary functions, Journal of the Association forComputing Machinery 21:4 (1974), pp. 586–589.

[5] Miklós Laczkovich, The removal of π from some undecidable problems involving elementary functions, Proc. Amer. Math.Soc. 131:7 (2003), pp. 2235–2240.

159.5 Further reading• Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron (1996). A = B. A. K. Peters. p. 212. ISBN 1-56881-063-6.

• Richardson, Daniel (1968). “Some undecidable problems involving elementary functions of a real variable”.Journal of Symbolic Logic 33 (4) (Association for Symbolic Logic). pp. 514–520. doi:10.2307/2271358.JSTOR 10.2307/2271358.

159.6 External links• Weisstein, Eric W., “Richardson’s theorem”, MathWorld.

Chapter 160

Robinson’s joint consistency theorem

Robinson’s joint consistency theorem is an important theorem of mathematical logic. It is related to Craig inter-polation and Beth definability.The classical formulation of Robinson’s joint consistency theorem is as follows:Let T1 and T2 be first-order theories. If T1 and T2 are consistent and the intersection T1 ∩ T2 is complete (in thecommon language of T1 and T2 ), then the union T1 ∪ T2 is consistent. Note that a theory is complete if it decidesevery formula, i.e. either T ⊢ φ or T ⊢ ¬φ .Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem:Let T1 and T2 be first-order theories. If T1 and T2 are consistent and if there is no formula φ in the common languageof T1 and T2 such that T1 ⊢ φ and T2 ⊢ ¬φ , then the union T1 ∪ T2 is consistent.

160.1 References• Boolos, George S.; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge Uni-versity Press. p. 264. ISBN 0-521-00758-5.

188

Chapter 161

Rowbottom cardinal

In set theory, a Rowbottom cardinal, introduced by Rowbottom (1971), is a certain kind of large cardinal number.An uncountable cardinal number κ is said to be Rowbottom if for every function f: [κ]<ω → λ (where λ < κ) there isa set H of order type κ that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsetsof H has countably many elements.Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, thetheories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.In general, Rowbottom cardinals need not be large cardinals in the usual sense: Rowbottom cardinals could besingular. It is an open question whether ZFC + “ ℵω is Rowbottom” is consistent. If it is, it has much higher con-sistency strength than the existence of a Rowbottom cardinal. The axiom of determinacy does imply that ℵω isRowbottom (but contradicts the axiom of choice).

161.1 References• Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd eded.). Springer. ISBN 3-540-00384-3.

• Rowbottom, Frederick (1971) [1964], “Some strong axioms of infinity incompatible with the axiom of con-structibility”,Annals of Pure and Applied Logic 3 (1): 1–44, doi:10.1016/0003-4843(71)90009-X, ISSN 0168-0072, MR 0323572

189

Chapter 162

Scattered order

This article is about order theory. For the Australian post-punk band, see Scattered Order.

In mathematical order theory, a scattered order is a linear order that contains no densely ordered subset with morethan one element (Harzheim 2005:193ff.)A characterization due to Hausdorff states that the class of all scattered orders is the smallest class of linear orderswhich contains the singleton orders and is closed under well-ordered and reverse well-ordered sums.Laver's theorem (generalizing Fraïssé's conjecture) states that the embedding relation on the class of countable unionsof scattered orders is a well-quasi-order (Harzheim 2005:265).The order topology of a scattered order is scattered. The converse implication does not hold, as witnessed by thelexicographic order on Q× Z .

162.1 References• Egbert Harzheim (2005). Ordered Sets. Springer. ISBN 0-387-24219-8.

• Laver, Richard (1971). “On Fraïssé's order type conjecture”. Annals ofMathematics 93 (1): 89–111. doi:10.2307/1970754.JSTOR 1970754.

190

Chapter 163

Semicomputable function

In computability theory, a semicomputable function is a partial function f : Q → R that can be approximatedeither from above or from below by a computable function.More precisely a partial function f : Q → R is upper semicomputable, meaning it can be approximated fromabove, if there exists a computable function ϕ(x, k) : Q× N → Q , where x is the desired parameter for f(x) andk is the level of approximation, such that:

• limk→∞ ϕ(x, k) = f(x)

• ∀k ∈ N : ϕ(x, k + 1) ≤ ϕ(x, k)

Completely analogous a partial function f : Q → R is lower semicomputable iff −f(x) is upper semicomputableor equivalently if there exists a computable function ϕ(x, k) such that

• limk→∞ ϕ(x, k) = f(x)

• ∀k ∈ N : ϕ(x, k + 1) ≥ ϕ(x, k)

If a partial function is both upper and lower semicomputable it is called computable.

163.1 See also• computability theory

163.2 References• Ming Li and Paul Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, pp 37–38, Springer,1997.

191

Chapter 164

Separating set

This article is about separating sets for functions. For use in graph theory, see connectivity (graph theory).

In mathematics a set of functions S from a set D to a set C is called a separating set for D or said to separate thepoints of D if for any two distinct elements x and y of D, there exists a function f in S so that f(x) ≠ f(y).[1]

Separating sets can be used to formulate a version of the Stone-Weierstrass theorem for real-valued functions on acompact Hausdorff space X, with the topology of uniform convergence. It states that any subalgebra of this space offunctions is dense if and only if it separates points. This is the version of the theorem originally proved by MarshallH. Stone.[1]

164.1 Examples• The singleton set consisting of the identity function on R separates the points of R.

• If X is a T1 normal topological space, then Urysohn’s lemma states that the set C(X) of continuous functionson X with real (or complex) values separates points on X.

164.2 References[1] Carothers, N. L. (2000), Real Analysis, Cambridge University Press, pp. 201–204, ISBN 9781139643160.

192

Chapter 165

Set constraint

Set contraints obtained from abstract interpretation of the Collatz algorithm.

In mathematics and theoretical computer science, a set constraint is an equation or an inequation between setsof terms. Similar to systems of (in)equations between numbers, methods are studied for solving systems of setconstraints. Different approaches admit different operators (like "∪", "∩", "\", and function application)[note 1] on setsand different (in)equation relations (like "=", "⊆", and "⊈") between set expressions.Systems of set constraints are useful to describe (in particular infinite) sets of ground terms.[note 2] They arise inprogram analysis, abstract interpretation, and type inference.

165.1 Relation to regular tree grammars

Each regular tree grammar can be systematically transformed into a system of set inclusions such that its minimalsolution corresponds to the tree language of the grammar.For example, the grammar (terminal and nonterminal symbols indicated by lower and upper case initials, respectively)with the rules

193

194 CHAPTER 165. SET CONSTRAINT

is transformed to the set inclusion system (constants and variables indicated by lower and upper case initials, respec-tively):

This system has a minimal solution, viz. ("L(N)" denoting the tree language corresponding to the nonterminal N inthe above tree grammar):

The maximal solution of the system is trivial; it assigns the set of all terms to every variable.

165.2 Literature• Aiken, A. (1995). Set Constraints: Results, Applications and Future Directions (Technical report). Univ. Berke-ley.

• Aiken, A., Kozen, D., Vardi, M., Wimmers, E.L. (May 1993). The Complexity of Set Constraints (Technicalreport). Computer Science Department, Cornell University. 93–1352.

• Aiken, A., Kozen, D., Vardi, M., Wimmers, E.L. (1994). “The Complexity of Set Constraints”. ComputerScience Logic'93. LNCS 832. Springer. pp. 1–17.

• Aiken, A., Wimmers, E.L. (1992). “Solving Systems of Set Constraints (Extended Abstract)". Seventh AnnualIEEE Symposium on Logic in Computer Science. pp. 329–340.

• Bachmair, Leo, Ganzinger, Harald, Waldmann, Uwe (1992). Set Constraints are the Monadic Class (Technicalreport). Max-Planck-Institut für Informatik. p. 13. MPI-I-92-240.

• Bachmair, Leo, Ganzinger, Harald, Waldmann, Uwe (1993). “Set Constraints are the Monadic Class”. EightAnnual IEEE Symposium on Logic in Computer Science. pp. 75–83.

• Charatonik, W. (Sep 1994). “Set Constraints in Some Equational Theories”. Proc. 1st Int. Conf. on Constraintsin Computational Logics (CCL). LNCS 845. Springer. pp. 304–319.

• Charatonik, Witold; Podelski, Andreas (2002). “Set Constraints with Intersection” (PDF). Information andComputation 179: 213–229. doi:10.1006/inco.2001.2952. Retrieved 11 May 2014.

• Charatonik, W., Podelski, A. (1998). Tobias Nipkow, ed. Co-definite Set Constraints. LNCS 1379. Springer-Verlag. pp. 211–225.

• Charatonik, W., Talbot, J.-M. (2002). Tison, S., ed. Atomic Set Constraints with Projection. LNCS 2378.Springer. pp. 311–325.

• Gilleron, R., Tison, S., Tommasi, M. (1993). “Solving Systems of Set Constraints using Tree Automata”. 10thAnnual Symposium on Theoretical Aspects of Computer Science. LNCS 665. Springer. pp. 505–514.

• Heintze, N., Jaffar, J. (1990). “A Decision Procedure for a Class of Set Constraints (Extended Abstract)".Fifth Annual IEEE Symposium on Logic in Computer Science. pp. 42–51.

• Heintze, N., Jaffar, J. (Feb 1991). A Decision Procedure for a Class of Set Constraints (Technical report).School of Computer Science, Carnegie Mellon University.

• Kozen, D. (1993). “Logical Aspects of Set Constraints”. Computer Science Logic'93 (PDF). LNCS 832. pp.175–188.

• Kozen, D. (1994). “Set Constraints and Logic Programming”. CCL. LNCS 845.

• Dexter Kozen (1998). “Set Constraints and Logic Programming” (PDF). Information and Computation 142:2–25. doi:10.1006/inco.1997.2694.

• Uribe, T.E. (1992). “Sorted Unification Using Set Constraints”. Proc. CADE–11. LNCS 607. pp. 163–177.

165.3. NOTES 195

165.2.1 Literature on negative constraints

• Aiken, A., Kozen, D., Wimmers, E.L. (Jun 1993). Decidability of Systems of Set Constraints with NegativeConstraints (Technical report). Computer Science Department, Cornell University. 93–1362.

• Charatonik, W., Pacholski, L. (Jul 1994). “Negative Set Constraints with Equality”. Ninth Annual IEEESymposium on Logic in Computer Science. pp. 128–136.

• R. Gilleron, S. Tison, M. Tommasi (1993). “Solving Systems of Set Constraints with Negated Subset Rela-tionships”. Proceedings of the 34th Symp. on Foundations of Computer Science. pp. 372–380.

• Gilleron, R., Tison, S., Tommasi, M. (1993). Solving Systems of Set Constraints with Negated Subset Relation-ships (Technical report). Laboratoire d'Informatique Fondamentale de Lille. IT 247.

• Stefansson, K. (Aug 1993). Systems of Set Constraints with Negative Constraints are NEXPTIME-Complete(Technical report). Computer Science Department, Cornell University. 93–1380.

• Stefansson, K. (1994). “Systems of Set Constraints with Negative Constraints are NEXPTIME-Complete”.Ninth Annual IEEE Symposium on Logic in Computer Science. pp. 137–141.

165.3 Notes[1] If f is an n-ary function symbol admitted in a term, then "f(E1,...,En)" is a set expression denoting the set f(t1,...,tn) :

t1∈E1 and ... and tn∈En , where E1,...,En are set expressions in turn.

[2] This is similar to describing e.g. a rational number as a solution to an equation a⋅x + b = 0, with integer coefficients a, b.

Chapter 166

Set function

In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input isa set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.

166.1 Examples

Examples of set functions include:

• The function that assigns to each set its cardinality, i.e. the number of members of the set, is a set function.

• The function

d(A) = limn→∞

|A ∩ 1, . . . , n|n

,

assigning densities to sufficiently well-behaved subsets A ⊆ 1, 2, 3, ..., is a set function.

• The Lebesgue measure is a set function that assigns a non-negative real number to each set of real numbers.(Kolmogorov and Fomin 1975)

• A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the emptyset is zero and the probability of the sample space is 1, with other sets given probabilities between 0 and 1.

• A possibility measure assigns a number between zero and one to each set in the powerset of some given set.See possibility theory.

• A Random set is a set-valued random variable. See Random compact set.

166.2 References• A.N. Kolmogorov and S.V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0

166.3 Further reading• Sobolev, V.I. (2001), “Set function”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• *Regular set function at Encyclopedia of Mathematics

196

Chapter 167

Shelah cardinal

In axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal κ is called Shelah iff for everyf : κ → κ , there exists a transitive class N and an elementary embedding j : V → N with critical point κ ; andVj(f)(κ) ⊂ N .A Shelah cardinal has a normal ultrafilter containing the set of weakly hyper-Woodin cardinals below it.

167.1 References• Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings ofthe American Mathematical Society 130/11, pp. 3385-3391, 2002, online

197

Chapter 168

Shrewd cardinal

In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by (Rathjen 1995)., ex-tending the definition of indescribable cardinals.A cardinal number κ is called λ-shrewd if for every proposition φ, and set A ⊆ Vκ with (Vκ₊λ, ∈, A) ⊧ φ there existsan α, λ' < κ with (Vα₊λ', ∈, A ∩ Vα) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ (including λ > κ).This definition extends the concept of indescribability to transfinite levels. A λ-shrewd cardinal is also μ-shrewd for anyordinal μ < λ. Shrewdness was developed by Michael Rathjen as part of his ordinal analysis of Π1

2-comprehension.It is essentially the nonrecursive analog to the stability property for admissible ordinals.More generally, a cardinal number κ is called λ-Π -shrewd if for every Π proposition φ, and set A ⊆ Vκ with (Vκ₊λ,∈, A) ⊧ φ there exists an α, λ' < κ with (Vα₊λ', ∈, A ∩ Vα) ⊧ φ.Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.For finite n, an n-Π -shrewd cardinals is the same thing as a Π n-indescribable cardinal.If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in κ. Rathjen does not state how shrewdcardinals compare to unfoldable cardinals, however.λ-shrewdness is an improved version of λ-indescribability, as defined in Drake; this cardinal property differs in thatthe reflected substructure must be (Vα₊λ, ∈, A ∩ Vα), making it impossible for a cardinal κ to be κ-indescribable.Also, the monotonicity property is lost: a λ-indescribable cardinal may fail to be α-indescribable for some ordinal α< λ.

168.1 References• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of

Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

• Rathjen, Michael (2006). “The Art of Ordinal Analysis” (PDF).

• Rathjen, Michael (1995), “Recent advances in ordinal analysis: Π12-CA and related systems”, The Bulletin of

Symbolic Logic 1 (4): 468–485, doi:10.2307/421132, ISSN 1079-8986, MR 1369172

198

Chapter 169

Soft set

Soft set theory is a generalization of fuzzy set theory, that was proposed byMolodtsov in 1999 to deal with uncertaintyin a parametric manner.[1] A soft set is a parametrised family of sets - intuitively, this is “soft” because the boundaryof the set depends on the parameters. Formally, a soft set, over a universal set X and set of parameters E is a pair (f,A) where A is a subset of E, and f is a function from A to the power set of X. For each e in A, the set f(e) is calledthe value set of e in (f, A).One of themost important steps for the new theory of soft sets was to definemappings on soft sets, which was achievedin 2009 by the mathematician Athar Kharal, with the results published in 2011.[2] Soft sets have also been appliedto the problem of medical diagnosis for use in medical expert systems. Fuzzy soft sets have also been introduced.Mappings on fuzzy soft sets were defined and studied by Kharal and Ahmad.[3]

169.1 Notes[1] Molodtsov, D. A. (1999). “Soft set theory—First results”. Computers & Mathematics With Applications 37 (4): 19–31.

doi:10.1016/S0898-1221(99)00056-5.

[2] Kharal, Athar; B. Ahmad (September 2011). “Mappings on Soft Classes”. New Mathematics and Natural Computation 7(3). doi:10.1142/S1793005711002025.

[3] Kharal, Athar; B. Ahmad (2009). “Mappings on Fuzzy Soft Classes”. Advances in Fuzzy Systems 2009. doi:10.1155/2009/407890.

169.2 References• Molodtsov D. A. A theory of soft sets. Moscow: Editorial URSS, 2004.

• Matsievsky S. V. Sets, multisets, fuzzy and soft sets without universe. Vestnik IKSUR, 2007, N. 10, pp. 44–52.

• Ahmad, B., Kharal, A.OnFuzzy Soft Sets. Advances in Fuzzy SystemsVolume 2009 (2009), Article ID 586507,6 pages doi:10.1155/2009/586507.

199

Chapter 170

Square principle

In mathematical set theory, the global square principle is a combinatorial principle introduced by Ronald Jensen inhis analysis of the fine structure of the constructible universe L.

170.1 Definition

Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system(Cβ)β∈Sing satisfying:

1. Cβ is a club set of β .

2. ot (Cβ) < β

3. If γ is a limit point of Cβ then γ ∈ Sing and Cγ = Cβ ∩ γ

170.2 Variant relative to a cardinal

Jensen introduced also a local version of the principle.[1] If κ is an uncountable cardinal, then κ asserts that thereis a sequence (Cβ | β of point limit a κ+) satisfying:

1. Cβ is a club set of β .

2. If cfβ < κ , then |Cβ | < κ

3. If γ is a limit point of Cβ then Cγ = Cβ ∩ γ

Jensen proved that this principle holds in the constructible universe for any uncountable cardinal κ.

170.3 Notes[1] Jech, Thomas (2003), Set Theory: Third Millennium Edition, Springer Monographs in Mathematics, Berlin, New York:

Springer-Verlag, ISBN 978-3-540-44085-7, p. 443.

• Jensen, R. Björn (1972), “The fine structure of the constructible hierarchy”, Annals of Mathematical Logic 4:229–308, doi:10.1016/0003-4843(72)90001-0, MR 0309729

200

Chapter 171

Strength (mathematical logic)

The relative strength of two systems of formal logic can be defined via model theory. Specifically, a logic α is saidto be as strong as a logic β if every elementary class in β is an elementary class in α .[1]

171.1 See also• Abstract logic

• Lindström’s theorem

171.2 References[1] Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors,Model-theoretic

logics, 1985 ISBN 0-387-90936-2 page 43

201

Chapter 172

Strong cardinal

In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.

172.1 Formal definition

If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j fromthe universe V into a transitive inner model M with critical point κ and

Vλ ⊆M

That is, M agrees with V on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ.

172.2 Relationship with other large cardinals

It is obvious from the definitions that strong cardinals lie below supercompact cardinals and above measurable car-dinals in the consistency strength hierarchy.They also lie below superstrong cardinals and Woodin cardinals. However, the least strong cardinal is larger than theleast superstrong cardinal.

172.3 References• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

202

Chapter 173

Strong partition cardinal

In Zermelo-Fraenkel set theory without the axiom of choice a strong partition cardinal is an uncountable well-ordered cardinal k such that every partition of the set [k]k of size k subsets of k into less than k pieces has ahomogeneous set of size k .The existence of strong partition cardinals contradicts the axiom of choice. The Axiom of determinacy implies thatℵ1 is a strong partition cardinal.

173.1 References• Henle, JamesM.; Kleinberg, EugeneM.; Watro, Ronald J. (1984), “On the ultrafilters and ultrapowers of strongpartition cardinals”, Journal of Symbolic Logic 49 (4): 1268–1272., doi:10.2307/2274277, JSTOR 2274277

• Apter, Arthur W.; Henle, James M.; Jackson, Stephen C. (1999), “The calculus of partition sequences, chang-ing cofinalities, and a question of Woodin”, Transactions of the American Mathematical Society 352 (3): 969–1003, doi:10.1090/S0002-9947-99-02554-4, JSTOR 118097, MR 1695015.

203

Chapter 174

Strongly compact cardinal

In mathematical set theory, a strongly compact cardinal is a certain kind of large cardinal number.A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ complete ultrafilter.Strongly compact cardinals were originally defined in terms of infinitary logic, where logical operators are allowed totake infinitely many operands. The logic on a regular cardinal κ is defined by requiring the number of operands foreach operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness propertyof finitary logic. Specifically, a statement which follows from some other collection of statements should also followfrom some subcollection having cardinality less than κ.The property of strong compactness may be weakened by only requiring this compactness property to hold when theoriginal collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. Acardinal is weakly compact if and only if it is κ-compact; this was the original definition of that concept.Strong compactness implies measurability, and is implied by supercompactness. Given that the relevant cardinalsexist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first stronglycompact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compactcardinals is strongly compact, but the least such limit is not supercompact.The consistency strength of strong compactness is strictly above that of a Woodin cardinal. Some set theorists con-jecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However,a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.Extendibility is a second-order analog of strong compactness.

174.1 References• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of

Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.

204

Chapter 175

Subcompact cardinal

In mathematics, a subcompact cardinal is a certain kind of large cardinal number.A cardinal number κ is subcompact if and only if for every A⊂H(κ+) there is a non-trivial elementary embeddingj:(H(μ+), B) → (H(κ+), A) with critical point μ and j(μ) = κ.Analogously, a cardinal number κ is quasicompact if and only if for every A⊂H(κ+) there is a non-trivial elementaryembedding j:(H(κ+), A) → (H(μ+), B) with critical point κ and j(κ) = μ.H(λ) consists of all sets whose transitive closure has cardinality less than λ.Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that itprojects large cardinal properties upwards. The relationship is analogous to that of extendible versus supercompactcardinals. Quasicompactness may be viewed as a strengthened or “boldface” version of 1-extendibility. Existenceof subcompact cardinals implies existence of many 1-extendible cardinals, and hence many superstrong cardinals.Existence of a 2κ-supercompact cardinal κ implies existence of many quasicompact cardinals.Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the Square Principle. If κ issubcompact, then the square principle fails at κ. Canonical inner models at the level of subcompact cardinals satisfythe square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in anycase the square principle can be forced for weaker cardinals.)Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models thatdo not use long extenders. For current inner models, the elementary embeddings included are determined by theireffect on P(κ) (as computed at the stage the embedding is included), where κ is the critical point. This prevents themfrom witnessing even a κ+ strongly compact cardinal κ.Subcompact and quasicompact cardinals were defined by Ronald Jensen.

175.1 See also• Hereditarily countable set

175.2 References• “Square in Core Models” in the September 2001 issue of the Bulletin of Symbolic Logic

205

Chapter 176

Subcountability

In constructive mathematics, a collection is subcountable if there exists a partial surjection from the natural numbersonto it. The name derives from the intuitive sense that such a collection is “no bigger” than the counting numbers.The concept is trivial in classical set theory, where a set is subcountable if and only if it is finite or countably infinite.Constructively it is consistent to assert the subcountability of some uncountable collections such as the real numbers.Indeed there are models of the constructive set theory CZF in which all sets are subcountable[1] and models of IZFin which all sets with apartness relations are subcountable.[2]

176.1 References[1] Rathjen, M. "Choice principles in constructive and classical set theories", Proceedings of the Logic Colloquium, 2002

[2] McCarty, J. "Subcountability under realizability", Notre Dame Journal of Formal Logic, Vol 27 no 2 April 1986

206

Chapter 177

Subtle cardinal

In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.A cardinal κ is called subtle if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for whichelement number δ (for an arbitrary δ), Aδ ⊂ δ there are α, β, belonging to C, with α<β, such that Aα=Aᵦ∩α. Acardinal κ is called ethereal if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for whichelement number δ (for an arbitrary δ), Aδ ⊂ δ and Aδ has the same cardinal as δ, there are α, β, belonging to C,with α<β, such that card(α)=card(Aᵦ∩Aα).Subtle cardinals were introduced by Jensen & Kunen (1969). Ethereal cardinals were introduced by Ketonen (1974).Any subtle cardinal is ethereal, and any strongly inaccessible ethereal cardinal is subtle.

177.1 Theorem

There is a subtle cardinal ≤κ if and only if every transitive set S of cardinality κ contains x and y such that x is aproper subset of y and x ≠ Ø and x ≠ Ø. An infinite ordinal κ is subtle if and only if for every λ<κ, every transitiveset S of cardinality κ includes a chain (under inclusion) of order type λ.

177.2 References• Friedman, Harvey (2001), “Subtle Cardinals and Linear Orderings”, Annals of Pure and Applied Logic 107(1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1

• Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V , Unpublished manuscript

• Ketonen, Jussi (1974), “Some combinatorial principles”, Transactions of the American Mathematical Society(Transactions of the American Mathematical Society, Vol. 188) 188: 387–394, doi:10.2307/1996785, ISSN0002-9947, JSTOR 1996785, MR 0332481

207

Chapter 178

Successor function

For other uses, see Successor.

In mathematics, the successor function or successor operation is a primitive recursive function S such that S(n) =n+1 for each natural number n. For example, S(1) = 2 and S(2) = 3.The successor function is used in the Peano axioms which define the natural numbers. As such, it is not defined byaddition, but rather is used to define all natural numbers beyond 0, as well as addition. For example, 1 is defined tobe S(0), and addition on natural numbers is defined recursively by:

This yields e.g. 5 + 2 = 5 + S(1) = S(5) + 1 = 6 + 1 = 6 + S(0) = S(6) + 0 = 7 + 0 = 7When natural numbers are constructed based on set theory, a common approach is to define the number 0 to be theempty set , and the successor S(x) to be x ∪ x . The axiom of infinity then guarantees the existence of a set ℕthat contains 0 and is closed with respect to S; members of ℕ are called natural numbers.[1]

The successor function is the level-0 foundation of the infinite hierarchy of hyperoperations (used to build addition,multiplication, exponentiation, tetration, etc.).It is also one of the primitive functions used in the characterization of computability by recursive functions.

178.1 See also• successor ordinal

• successor cardinal

178.2 References

Paul R. Halmos (1968). Naive Set Theory. Nostrand.

[1] Halmos, Chapter 11

208

Chapter 179

Sudan function

In the theory of computation, the Sudan function is an example of a function that is recursive, but not primitiverecursive. This is also true of the better-known Ackermann function. The Sudan function was the first function havingthis property to be published.It was discovered (and published[1]) in 1927 by Gabriel Sudan, a Romanian mathematician who was a student ofDavid Hilbert.

179.1 Definition

F0(x, y) = x+ y,

Fn+1(x, 0) = x, n ≥ 0

Fn+1(x, y + 1) = Fn(Fn+1(x, y), Fn+1(x, y) + y + 1), n ≥ 0.

179.2 Value Tables

In general, F1(x, y) is equal to F1(0, y) + 2y x.

179.3 References• Cristian Calude, Solomon Marcus, Ionel Tevy, The first example of a recursive function which is not primitive

recursive, Historia Mathematica 6 (1979), no. 4, 380–384 doi:10.1016/0315-0860(79)90024-7

[1] Bull. Math. Soc. Roumaine Sci. 30 (1927), 11 - 30; Jbuch 53, 171

209

Chapter 180

Supernatural number

In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers,are a generalization of the natural numbers. They were used by Ernst Steinitz[1] in 1910 as a part of his work on fieldtheory.A supernatural number ω is a formal product:

ω =∏p

pnp ,

where p runs over all prime numbers, and each np is zero, a natural number or infinity. Sometimes vp(ω) is usedinstead of np . If no np = ∞ and there are only a finite number of non-zero np then we recover the positiveintegers. Slightly less intuitively, if all np are∞ , we get zero. Supernatural numbers extend beyond natural numbersby allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide ω “infinitelyoften,” by taking that prime’s corresponding exponent to be the symbol∞ .There is no natural way to add supernatural numbers, but they can bemultiplied, with

∏p p

np ·∏

p pmp =

∏p p

np+mp

. Similarly, the notion of divisibility extends to the supernaturals with ω1 | ω2 if vp(ω1) ≤ vp(ω2) for all p . Thenotion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers,by defining

lcm(ωi) =∏p

psup(vp(ωi))

gcd(ωi) =∏p

pinf(vp(ωi))

With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernaturalnumber. We can also extend the usual p -adic order functions to supernatural numbers by defining vp(ω) = np foreach pSupernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case manyof the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of afinite field.[2] They are also used implicitly in many number-theoretical proofs, such as the density of the square-freeintegers and bounds for odd perfect numbers.

180.1 See also

• profinite integer

210

180.2. REFERENCES 211

180.2 References[1] Steinitz, Ernst (1910). “Algebraische Theorie der Körper”. Journal für die reine und angewandte Mathematik: 167–309.

ISSN 0075-4102. JFM 41.0445.03.

[2] Brawley & Schnibben (1989) pp.25-26

• Brawley, Joel V.; Schnibben, George E. (1989). Infinite algebraic extensions of finite fields. ContemporaryMathematics 95. Providence, RI: American Mathematical Society. pp. 23–26. ISBN 0-8218-5101-2. Zbl0674.12009.

• Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs 124.Providence, RI: American Mathematical Society. p. 125. ISBN 0-8218-4041-X. Zbl 1103.12002.

• Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd ed.). Springer-Verlag. p. 520. ISBN 978-3-540-77269-9. Zbl 1145.12001.

180.3 External links• Planet Math: Supernatural number

Chapter 181

Superposition calculus

The superposition calculus is a calculus for reasoning in equational first-order logic. It has been developed in theearly 1990s and combines concepts from first-order resolution with ordering-based equality handling as developedin the context of (unfailing) Knuth–Bendix completion. It can be seen as a generalization of either resolution (toequational logic) or unfailing completion (to full clausal logic). As most first-order calculi, superposition tries to showthe unsatisfiability of a set of first-order clauses, i.e. it performs proofs by refutation. Superposition is refutation-complete — given unlimited resources and a fair derivation strategy, from any unsatisfiable clause set a contradictionwill eventually be derived.As of 2007, most of the (state-of-the-art) theorem provers for first-order logic are based on superposition (e.g. the Eequational theorem prover), although only a few implement the pure calculus.

181.1 Implementations• E

• SPASS

• Vampire

• Waldmeister

• Ayane at Google Code

181.2 References• Rewrite-Based Equational TheoremProvingwith Selection and Simplification, Leo Bachmair andHaraldGanzinger,Journal of Logic and Computation 3(4), 1994.

• Paramodulation-Based Theorem Proving, Robert Nieuwenhuis and Alberto Rubio, Handbook of AutomatedReasoning I(7), Elsevier Science and MIT Press, 2001.

212

Chapter 182

Superstrong cardinal

In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding j : V→ M from V into a transitive inner model M with critical point κ and Vj(κ) ⊆ M.Similarly, a cardinal κ is n-superstrong if and only if there exists an elementary embedding j : V →M from V intoa transitive inner model M with critical point κ and Vjn(κ) ⊆ M. Akihiro Kanamori has shown that the consistencystrength of an n+1-superstrong cardinal exceeds that of an n-huge cardinal for each n > 0.

182.1 References• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

213

Chapter 183

Suslin cardinal

In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is λ-Suslin but P is notλ'-Suslin for any λ' < λ. It is named after the Russian mathematician Mikhail Yakovlevich Suslin (1894–1919).

183.1 See also• Suslin representation

• Suslin line

• AD+

183.2 References• Howard Becker, The restriction of a Borel equivalence relation to a sparse set, Arch. Math. Logic 42, 335–347(2003), doi:10.1007/s001530200142

214

Chapter 184

Suslin representation

In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whoseprojection is that set of reals. More generally, a subset A of κω is λ-Suslin if there is a tree T on κ × λ such that A =p[T].By a tree on κ × λ we mean here a subset T of the union of κi × λi for all i ∈ N (or i < ω in set-theoretical notation).Here, p[T] = f | ∃g : (f,g) ∈ [T] is the projection of T, where [T] = (f, g ) | ∀n ∈ ω : (f(n), g(n) ∈ T is the setof branches through T.Since [T] is a closed set for the product topology on κω × λω where κ and λ are equipped with the discrete topology(and all closed sets in κω × λω come in this way from some tree on κ × λ), λ-Suslin subsets of κω are projections ofclosed subsets in κω × λω.When one talks of Suslin sets without specifying the space, then one usually means Suslin subsets of R, which de-scriptive set theorists usually take to be the set ωω.

184.1 See also• Suslin cardinal

• Suslin operation

184.2 External links• R. Ketchersid, The strength of an ω1-dense ideal on ω1 under CH, 2004.

215

Chapter 185

Suslin tree

In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable.(An antichain is a set of elements such that any two are incomparable.) They are named after Mikhail YakovlevichSuslin.Every Suslin tree is an Aronszajn tree.The existence of a Suslin tree is logically independent of ZFC, and is equivalent to the existence of a Suslin line(shown by Kurepa (1935)) or a Suslin algebra. The diamond principle, a consequence of V=L, implies that there isa Suslin tree, and Martin’s axiom MA(ℵ1) implies that there are no Suslin trees.More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain hascardinality less than κ. In particular a Suslin tree is the same as a ω1-Suslin tree. Jensen (1972) showed that if V=Lthen there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the Generalized Continuum Hypothesisimplies the existence of an ℵ2-Suslin tree, is a longstanding open problem.

185.1 See also• Kurepa tree

• Suslin’s problem

185.2 References• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics,Springer, ISBN3-540-44085-2

• Jensen, R. Björn (1972), “The fine structure of the constructible hierarchy.”, Ann. Math. Logic 4 (3): 229–308,doi:10.1016/0003-4843(72)90001-0, MR 0309729 erratum, ibid. 4 (1972), 443.

• Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001

• Kurepa, G. (1935), “Ensembles ordonnés et ramifiés”, Publ. math. Univ. Belgrade 4: 1–138, JFM 61.0980.01,Zbl 0014.39401

216

Chapter 186

Switching circuit theory

Switching circuit theory is the mathematical study of the properties of networks of idealized switches.Such networks may be strictly combinational logic, in which their output state is only a function of the present stateof their inputs; or may also contain sequential elements, where the present state depends on the present state and paststates; in that sense, sequential circuits are said to include “memory” of past states. An important class of sequentialcircuits are state machines. Switching circuit theory is applicable to the design of telephone systems, computers, andsimilar systems.In the paper A Symbolic Analysis of Relay and Switching Circuits of 1938, Claude Shannon showed that the two-valued Boolean algebra can describe the operation of switching circuits. The principles of Boolean algebra are appliedto switches, providing mathematical tools for analysis and synthesis of any switching system.Ideal switches are considered as having only two exclusive states, for example, open or closed. In some analysis, thestate of a switch can be considered to have no influence on the output of the system and is designated as a “don't care”state. In complex networks it is necessary to also account for the finite switching time of physical switches; wheretwo or more different paths in a network may affect the output, these delays may result in a “logic hazard” or "racecondition" where the output state changes due to the different propagation times through the network.

186.1 See also• Karnaugh map

• Boolean circuit

• C-element

• Circuit minimization

• Circuit complexity

• Circuit switching

• Logic design

• Logic in computer science

• Logic gate

• Nonblocking minimal spanning switch

• Quine–McCluskey algorithm

• Relay - the kind of logic device Shannon was concerned with in 1938

• Programmable logic controller - computer software mimics relay circuits for industrial applications

• Switching lemma

• Unate function

217

218 CHAPTER 186. SWITCHING CIRCUIT THEORY

186.2 References• Keister, William; Ritchie, Alistair E.; Washburn, Seth H. (1963) [1951]. The Design of Switching Circuits.The Bell Telephone Laboratories Series. Princeton, NJ: D. Van Nostrand Company.

• Caldwell, Samuel H. (1965) [1958]. Switching Circuits and Logical Design. New York: John Wiley & Sons.

• Shannon, C. E. (1938). “A Symbolic Analysis of Relay and Switching Circuits”. Trans. AIEE 57 (12): 713–723. doi:10.1109/T-AIEE.1938.5057767.

Chapter 187

Symmetric set

In mathematics, a nonempty subset S of a group G is said to be symmetric if

S = S−1

where S−1 = x−1 : x ∈ S . In other words, S is symmetric if x−1 ∈ S whenever x ∈ S .If S is a subset of a vector space, then S is said to be symmetric if it is symmetric with respect to the additive groupstructure of the vector space; that is, if S = −S = −x : x ∈ S .

187.1 Examples• In R, examples of symmetric sets are intervals of the type (−k, k) with k > 0 , and the sets Z and −1, 1 .

• Any vector subspace in a vector space is a symmetric set.

• If S is any subset of a group, then SS−1 and S−1S are symmetric sets.

187.2 References• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.

• W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

This article incorporates material from symmetric set on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

219

Chapter 188

Systems of Logic Based on Ordinals

Systems of Logic Based on Ordinals was the PhD dissertation of the mathematician Alan Turing.[1][2]

The thesis is an exploration of formal mathematical systems after Gödel’s theorem. Gödel showed for that any formalsystem S powerful enough to represent arithmetic, there is a theorem G which is true but the system is unable toprove. G could be added as an additional axiom to the system in place of a proof. However this would create anew system S' with its own unprovable true theorem G', and so on. Turing’s thesis considers iterating the process toinfinity, creating a system with an infinite set of axioms.The thesis was completed at Princeton under Alonzo Church and was a classic work in mathematics which introducedthe concept of ordinal logic.[3]

Martin Davis states that although Turing’s use of a computing oracle is not a major focus of the dissertation, it hasproven to be highly influential in theoretical computer science, e.g. in the polynomial time hierarchy.[4]

188.1 References[1] Turing, Alan (1938). Systems of Logic Based on Ordinals (PhD thesis). Princeton University. doi:10.1112/plms/s2-

45.1.161.

[2] Turing, A. M. (1939). “Systems of Logic Based on Ordinals”. Proceedings of the London Mathematical Society: 161–228.doi:10.1112/plms/s2-45.1.161.

[3] Solomon Feferman, Turing in the Land of O(z) in “The universal Turing machine: a half-century survey” by Rolf Herken1995 ISBN 3-211-82637-8 page 111

[4] Martin Davis “Computability, Computation and the Real World”, in Imagination and Rigor edited by Settimo Termini2006 ISBN 88-470-0320-2 pages 63-66

188.2 External links• “Turing’s Princeton Dissertation”. Princeton University Press. Retrieved January 10, 2012.

• Solomon Feferman (November 2006), “Turing’s Thesis” (PDF), Notices of the AMS 53 (10)

220

Chapter 189

Tail sequence

In mathematics, specifically set theory, a tail sequence is an unbounded sequence of contiguous ordinals. Formally,let β be a limit ordinal. Then a γ-sequence s ≡ ⟨sα|α < γ⟩ is a tail sequence in β if there exists an ε < β such that sis a normal sequence assuming all values in β \ ϵ.

221

Chapter 190

Takeuti’s conjecture

In mathematics, Takeuti’s conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-orderlogic has cut-elimination (Takeuti 1953). It was settled positively:

• By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966);

• Independently by Takahashi by a similar technique (Takahashi 1967);

• It is a corollary of Jean-Yves Girard's syntactic proof of strong normalization for System F.

Takeuti’s conjecture is equivalent to the consistency of second-order arithmetic in the sense that each of the statementscan be derived from each other in the weak system PRA of arithmetic; consistency refers here to the truth of the Gödelsentence for second-order arithmetic. It is also equivalent to the strong normalization of the Girard/Reynold’s SystemF.

190.1 See also• Hilbert’s second problem

190.2 References• William W. Tait, 1966. A nonconstructive proof of Gentzen's Hauptsatz for second order predicate logic. In

Bulletin of the American Mathematical Society, 72:980–983.

• Gaisi Takeuti, 1953. On a generalized logic calculus. In Japanese Journal of Mathematics, 23:39–96. Anerrata to this article was published in the same journal, 24:149–156, 1954.

• Moto-o Takahashi, 1967. A proof of cut-elimination in simple type theory. In Japanese Mathematical Society,10:44–45.

222

Chapter 191

Tarski–Kuratowski algorithm

In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithmwhich provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy.The algorithm is named after Alfred Tarski and Kazimierz Kuratowski.

191.1 Algorithm

The Tarski–Kuratowski algorithm for the arithmetical hierarchy:

1. Convert the formula to prenex normal form.

2. If the formula is quantifier-free, it is in Σ00 and Π0

0 .

3. Otherwise, count the number of alternations of quantifiers; call this k.

4. If the first quantifier is ∃, the formula is in Σ0k+1 .

5. If the first quantifier is ∀, the formula is in Π0k+1 .

191.2 References• Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1;ISBN 0-07-053522-1

223

Chapter 192

Tav (number)

In his work on set theory, Georg Cantor denoted the collection of all cardinal numbers by the last letter of the Hebrewalphabet, ת (transliterated as Taf, Tav, or Taw.) As Cantor realized, this collection could not itself have a cardinality,as this would lead to a paradox of the Burali-Forti type. Cantor instead said that it was an “inconsistent” collectionwhich was absolutely infinite.[1][4]

192.1 See also• Taw (letter)

• Aleph number

• Absolute Infinite

192.2 References[1] Gesammelte Abhandlungen,[2] Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962,

pp. 443–447; translated into English in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jeanvan Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purportto be a letter from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan-Guinness has discovered,[3] this isin fact an amalgamation by Cantor’s editor, Ernst Zermelo, of two letters from Cantor to Dedekind, the first dated July 28and the second dated August 3.

[2] Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Georg Cantor, ed. Ernst Zermelo, with biographyby Adolf Fraenkel; orig. pub. Berlin: Verlag von Julius Springer, 1932; reprinted Hildesheim: Georg Olms, 1962, andBerlin: Springer-Verlag, 1980, ISBN 3-540-09849-6.

[3] TheRediscovery of the Cantor-DedekindCorrespondence, I. Grattan-Guinness, Jahresbericht der DeutschenMathematiker-Vereinigung 76 (1974/75), pp. 104–139, at p. 126 ff.

[4] TheCorrespondence betweenGeorgCantor and Philip Jourdain, I. Grattan-Guinness, Jahresbericht der DeutschenMathematiker-Vereinigung 73 (1971/72), pp. 111–130, at pp. 116–117.

224

Chapter 193

Teichmüller–Tukey lemma

In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey’s lemma), named after John Tukeyand Oswald Teichmüller, states that every nonempty collection of finite character has a maximal element with respectto inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equivalent to the axiom of choice,and therefore to the well-ordering theorem, Zorn’s lemma, and the Hausdorff maximal principle.[1]

193.1 Definitions

A family of sets is of finite character provided it has the following properties:

1. For each A ∈ F , every finite subset of A belongs to F .

2. If every finite subset of a given set A belongs to F , then A belongs to F .

193.2 Statement of the Lemma

Whenever F ⊆ P(A) is of finite character and X ∈ F , there is a maximal Y ∈ F such that X ⊆ Y .[2]

193.3 Applications

In linear algebra, the lemma may be used to show the existence of a basis. Let V be a vector space. Consider thecollectionF of linearly independent sets of vectors. This is a collection of finite character Thus, a maximal set exists,which must then span V and be a basis for V.

193.4 Notes[1] Jech, Thomas J. (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 978-0-486-46624-8.

[2] Kunen, Kenneth (2009). The Foundations of Mathematics. College Publications. ISBN 978-1-904987-14-7.

193.5 References• Brillinger, David R. “John Wilder Tukey”

225

Chapter 194

The Paradoxes of the Infinite

The Paradoxes of the Infinite (German title: Paradoxien des Unendlichen) is a mathematical work by BernardBolzano on the theory of sets. It was published by a friend in 1851, three years after Bolzano’s death. The workcontained many interesting results in set theory. Bolzano expanded on the theme of Galileo’s paradox, giving moreexamples of correspondences between the elements of an infinite set and proper subsets of infinite sets. In the workhe also coined the term Menge, rendered in English as “set”.

194.1 References• Paradoxes of the Infinite; trans. by D.A.Steele; London: Routledge, 1950

• Bolzano, Bernard (1851), Paradoxien des Unendlichen (PDF), C.H. Reclam (German original) Faksimile

• Burton, David (1997), The History of Mathematics: An Introduction (Third ed.), McGraw-Hill, p. 592

226

Chapter 195

Theory of pure equality

In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only theequality relation symbol, and includes no non-logical axioms at all (Monk 1976:240–242). This theory is consistent,as any set with the usual equality relation provides an interpretation.The theory of pure equality was proven to be decidable by Löwenheim in 1915. If an additional axiom is addedsaying either that there are exactly m objects, for a fixed natural number m, or an axiom scheme is added stating thereare infinitely many objects, the resulting theory is complete.

195.1 References• Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1

227

Chapter 196

Trichotomy (mathematics)

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.[1] Moregenerally, trichotomy is the property of an order relation < on a set X that for any x and y, exactly one of thefollowing holds: x < y, x = y , or x > y .In mathematical notation, this is

∀x ∈ X ∀y ∈ X ((x < y ∧¬(y < x)∧¬(x = y) )∨ (¬(x < y)∧ y < x∧¬(x = y) )∨ (¬(x < y)∧¬(y < x)∧x = y )) .

Assuming that the ordering is irreflexive and transitive, this can be simplified to

∀x ∈ X ∀y ∈ X ((x < y) ∨ (y < x) ∨ (x = y)) .

In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore alsofor comparisons between integers and between rational numbers. The law does not hold in general in intuitionisticlogic.In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbersof well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds betweenarbitrary cardinal numbers (because they are all well-orderable in that case).[2]

More generally, a binary relation R on X is trichotomous if for all x and y in X exactly one of xRy, yRx or x=y holds.If such a relation is also transitive it is a strict total order; this is a special case of a strict weak order. For example, inthe case of three element set a,b,c the relation R given by aRb, aRc, bRc is a strict total order, while the relation Rgiven by the cyclic aRb, bRc, cRa is a non-transitive trichotomous relation.In the definition of an ordered integral domain or ordered field, the law of trichotomy is usually taken as morefoundational than the law of total order.A trichotomous relation cannot be reflexive, since xRx must be false. If a trichotomous relation is transitive, it istrivially antisymmetric and also asymmetric, since xRy and yRx cannot both hold.

196.1 See also• Dichotomy

• Law of noncontradiction

• Law of excluded middle

196.2 References[1] http://mathworld.wolfram.com/TrichotomyLaw.html

228

196.2. REFERENCES 229

[2] Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.

Chapter 197

Truth-table reduction

In computability theory, a truth-table reduction is a reduction from one set of natural numbers to another. As a“tool”, it is weaker than Turing reduction, since not every Turing reduction between sets can be performed by a truth-table reduction, but every truth-table reduction can be performed by a Turing reduction. For the same reason it issaid to be a stronger reducibility than Turing reducibility, because it implies Turing reducibility. Aweak truth-tablereduction is a related type of reduction which is so named because it weakens the constraints placed on a truth-tablereduction, and provides a weaker equivalence classification; as such, a “weak truth-table reduction” can actually bemore powerful than a truth-table reduction as a “tool”, and perform a reduction which is not performable by truthtable.A Turing reduction from a set B to a set A computes the membership of a single element in B by asking questionsabout the membership of various elements in A during the computation; it may adaptively determine which questionsit asks based upon answers to previous questions. In contrast, a truth-table reduction or a weak truth-table reductionmust present all of its (finitely many) oracle queries at the same time. In a truth-table reduction, the reduction alsogives a boolean function (a truth table) which, when given the answers to the queries, will produce the final answer ofthe reduction. In a weak truth-table reduction, the reduction uses the oracle answers as a basis for further computationwhich may depend on the given answers but may not ask further questions of the oracle.Equivalently, a weak truth-table reduction is a Turing reduction for which the use of the reduction is bounded by acomputable function. For this reason, they are sometimes referred to as bounded Turing (bT) reductions rather thanas weak truth-table (wtt) reductions.

197.1 Properties

As every truth-table reduction is a Turing reduction, if A is truth-table reducible to B (A ≤ B), then A is alsoTuring reducible to B (A ≤T B). Considering also one-one reducibility, many-one reducibility and weak truth-tablereducibility,

A ≤1 B ⇒ A ≤m B ⇒ A ≤tt B ⇒ A ≤wtt B ⇒ A ≤T B

or in other words, one-one reducibility implies many-one reducibility, which implies truth-table reducibility, whichin turn implies weak truth-table reducibility, which in turn implies Turing reducibility.

197.2 References• H. Rogers, Jr., 1967. The Theory of Recursive Functions and Effective Computability, second edition 1987,MIT Press. ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1

230

Chapter 198

Ulam matrix

In mathematical set theory, anUlammatrix is an array of subsets of a cardinal number with certain properties. Ulammatrices were introduced by Ulam (1930) in his work on measurable cardinals: they may be used, for example, toshow that a real-valued measurable cardinal is weakly inaccessible.[1]

198.1 Definition

Suppose that κ and λ are cardinal numbers, and let F be a λ-complete filter on λ. An Ulam matrix is a collection ofsubsets Aαᵦ of λ indexed by α in κ, β in λ such that

• If β is not γ then Aαᵦ and Aαᵧ are disjoint.

• For each β the union of the sets Aαᵦ is in the filter F.

198.2 References[1] Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York:

Springer-Verlag, p. 131, ISBN 978-3-540-44085-7, Zbl 1007.03002

• Ulam, Stanisław (1930), “Zur Masstheorie in der allgemeinen Mengenlehre”, Fundamenta Mathematicae 16(1): 140–150

231

Chapter 199

Unfoldable cardinal

In mathematics, an unfoldable cardinal is a certain kind of large cardinal number.Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivialelementary embedding j of M into a transitive model with the critical point of j being κ and j(κ) ≥ λ.A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ.A cardinal number κ is strongly λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivialelementary embedding j of M into a transitive model “N” with the critical point of j being κ, j(κ) ≥ λ, and V(λ) is asubset of N. Without loss of generality, we can demand also that N contains all its sequences of length λ.Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ.These properties are essentially weaker versions of strong and supercompact cardinals, consistent with V = L. Manytheorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. Forexample, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the properforcing axiom.A Ramsey cardinal is unfoldable, and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V,however.In L, any unfoldable cardinal is strongly unfoldable; thus unfoldables and strongly unfoldables have the same consistencystrength.A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact. A κ+ω-unfoldablecardinal is totally indescribable and preceded by a stationary set of totally indescribable cardinals.

199.1 References• Unfoldable Cardinals and the GCH, Joel David Hamkins. The Journal of Symbolic Logic, Vol. 66, No. 3(Sep., 2001), pp. 1186–1198 doi:10.2307/2695100

• Strongly unfoldable cardinals made indestructible, Thomas A. Johnstone. J. Symbolic Logic, Volume 73, Issue4 (2008), 1215-1248. doi:10.2178/jsl/1230396915

• Diamond (on the regulars) can fail at any strongly unfoldable cardinal, Joel DavidHamkins (TheCity Universityof New York), Mirna Džamonja (University of East Anglia). (Submitted to arxiv (http://arxiv.org/abs/math/0409304) on 17 Sep 2004)

232

Chapter 200

Universally Baire set

In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire spaceor Cantor space) is called universally Baire if it has a certain strong regularity property. Universally Baire sets playan important role in Ω-logic, a very strong logical system invented by W. Hugh Woodin and the centerpiece of hisargument against the continuum hypothesis of Georg Cantor.

200.1 Definition

A subset A of the Baire space is universally Baire if it has one of the following equivalent properties:

1. For every notion of forcing, there are trees T and U such that A is the projection of the set of all branchesthrough T, and it is forced that the projections of the branches through T and the branches through U arecomplements of each other.

2. For every compact Hausdorff space Ω, and every continuous function f fromΩ to the Baire space, the preimageof A under f has the property of Baire in Ω.

3. For every cardinal λ and every continuous function f from λω to the Baire space, the preimage of A under fhas the property of Baire.

200.2 References• Bagaria, Joan; Todorcevic, Stevo (eds.). Set Theory: Centre de Recerca Matemàtica Barcelona, 2003-2004.Trends in Mathematics. ISBN 978-3-7643-7691-8.

• Feng, Qi; Magidor, Menachem; Woodin, Hugh. Judah, H.; Just, W.; Woodin, Hugh, eds. Set Theory of theContinuum. Mathematical Sciences Research Institute Publications.

233

Chapter 201

Vopěnka’s principle

In mathematics, Vopěnka’s principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarityformalized through elementary embeddings.Vopěnka’s principle was first introduced by Petr Vopěnka and independently considered by Keisler, and was written upby Solovay, Reinhardt & Kanamori (1978). According to Pudlák (2013, p. 204), Vopěnka’s principle was originallyintended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle asa bogus large cardinal property, planning to show later that it was not consistent. However before publishing hisinconsistency proof he found a flaw in it.

201.1 Definition

Vopěnka’s principle asserts that for every proper class of binary relations (each with set-sized domain), there is oneelementarily embeddable into another. This cannot be stated as a single sentence of ZFC as it involves a quantificationover classes. A cardinal κ is called a Vopěnka cardinal if it is inaccessible and Vopěnka’s principle holds in the rankVκ (allowing arbitrary S ⊂ Vκ as “classes”). [1]

Many equivalent formulations are possible. For example, Vopěnka’s principle is equivalent to each of the followingstatements.

• For every proper class of simple directed graphs, there are two members of the class with a homomorphismbetween them.[2]

• For any signatureΣ and any proper class ofΣ-structures, there are twomembers of the class with an elementaryembedding between them.[1][2]

• For every predicate P and proper class S of ordinals, there is a non-trivial elementary embedding j:(Vκ, ∈, P)→ (Vλ, ∈, P) for some κ and λ in S.[1]

• The category of ordinals cannot be fully embedded in the category of graphs.[2]

• Every subfunctor of an accessible functor is accessible.[2]

• (In a definable classes setting) For every natural number n, there exists a C(n)-extendible cardinal.[3]

201.2 Strength

Even when restricted to predicates and proper classes definable in first order set theory, the principle implies existenceof Σ correct extendible cardinals for every n.If κ is an almost huge cardinal, then a strong form of Vopenka’s principle holds in Vκ:

There is a κ-complete ultrafilter U such that for every Ri: i < κ where each Ri is a binary relation andRi ∈ Vκ, there is S ∈ U and a non-trivial elementary embedding j: Ra→ Rb for every a < b in S.

234

201.3. REFERENCES 235

201.3 References[1] Kanamori, Akihiro (2003). The higher infinite : large cardinals in set theory from their beginnings (2nd ed.). Berlin [u.a.]:

Springer. ISBN 9783540003847.

[2] Rosicky, Jiří Adámek ; Jiří (1994). Locally presentable and accessible categories (Digital print. 2004. ed.). Cambridge[u.a.]: Cambridge Univ. Press. ISBN 0521422612.

[3] Bagaria, Joan (23December 2011). "C(n)-cardinals”. Archive forMathematical Logic 51 (3-4): 213–240. doi:10.1007/s00153-011-0261-8.

• Kanamori, A. (1978), “On Vopěnka’s and related principles”, Logic Colloquium '77 (Proc. Conf., Wrocław,1977), Stud. Logic Foundations Math. 96, Amsterdam-New York: North-Holland, pp. 145–153, ISBN 0-444-85178-X, MR 0519809

• Pudlák, Pavel (2013), Logical foundations of mathematics and computational complexity. A gentle introduction,Springer Monographs in Mathematics, Springer, ISBN 978-3-319-00118-0, MR 3076860

• Solovay, Robert M.; Reinhardt, WilliamN.; Kanamori, Akihiro (1978), “Strong axioms of infinity and elemen-tary embeddings” (PDF), Annals of Mathematical Logic 13 (1): 73–116, doi:10.1016/0003-4843(78)90031-1

201.4 External links

Friedman, HarveyM. (2005), EMBEDDINGAXIOMS gives a number of equivalent definitions of Vopěnka’s principle.

Chapter 202

Zero dagger

In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovayin unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on somebrowsers.) The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions.Specifically, if ZFC is consistent, then ZFC + “0† does not exist” is consistent. ZFC + “0† exists” is not known to beinconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (seelarge cardinal for a discussion). It is usually formulated as follows:

0† exists if and only if there exists a non-trivial elementary embedding j : L[U] → L[U] for the rela-tivized Gödel constructible universe L[U], where U is an ultrafilter witnessing that some cardinal κ ismeasurable.

If 0† exists, then a careful analysis of the embeddings of L[U] into itself reveals that there is a closed unboundedsubset of κ, and a closed unbounded proper class of ordinals greater than κ, which together are indiscernible for thestructure (L,∈, U) , and 0† is defined to be the set of Gödel numbers of the true formulas about the indiscerniblesin L[U].Solovay showed that the existence of 0† follows from the existence of two measurable cardinals. It is traditionallyconsidered a large cardinal axiom, although it is not a large cardinal, nor indeed a cardinal at all.

202.1 See also• 0#: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.

202.2 References• Kanamori, Akihiro; Awerbuch-Friedlander, Tamara (1990). “The compleat 0†". Zeitschrift für Mathematische

Logik und Grundlagen der Mathematik 36 (2): 133–141. doi:10.1002/malq.19900360206. ISSN 0044-3050.MR 1068949

• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nded.). Springer. ISBN 3-540-00384-3.

202.3 External links• Definition by “Zentralblatt math database” (PDF)

236

Chapter 203

Łoś–Tarski preservation theorem

The Łoś–Tarski theorem is a theorem in model theory, a branch of mathematics, that states that the set of formu-las preserved under taking substructures is exactly the set of universal formulas (Hodges 1997). The theorem wasdiscovered by Jerzy Łoś and Alfred Tarski.

203.1 Statement

Let T be a theory in a first-order language L and Φ(x) a set of formulas of L . (The set of sequence of variables xneed not be finite.) Then the following are equivalent:

1. If A andB are models of T , A ⊆ B , a is a sequence of elements of A andB |=∧Φ(a) , then A |=

∧Φ(a)

.( Φ is preserved in substructures for models of T )

2. Φ is equivalent modulo T to a set Ψ(x) of ∀1 formulas of L .

A formula is ∀1 if and only if it is of the form ∀x[ψ(x)] where ψ(x) is quantifier-free.Note that this property fails for finite models.

203.2 References• Peter G. Hinman (2005), Fundamentals of Mathematical Logic, A K Peters, ISBN 1568812620.

• Hodges (1997), A Shorter Model Theory, Cambridge University Press, ISBN 0521587131.

237

Chapter 204

Θ (set theory)

In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection fromthe reals onto α.If the axiom of choice (AC) holds (or even if the reals can be wellordered) then Θ is simply (2ℵ0)+ , the cardinalsuccessor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choicefails, such as models of the axiom of determinacy.Θ is also the supremum of the lengths of all prewellorderings of the reals.

204.1 Proof of existence

It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto whichthere is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinalsare wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the setof all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals intothe set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powersetaxiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, bythe Burali-Forti paradox.

238

Chapter 205

Ψ₀(Ωω)

The correct title of this article isΨ0(Ωω). It appears incorrectly here because of technical restrictions.

In mathematics, Ψ0(Ωω) is a large countable ordinal that is used to measure the proof-theoretic strength of somemathematical systems. In particular, it is the proof theoretic ordinal of the subsystem Π1

1 -CA0 of second-orderarithmetic; this is one of the “big five” subsystems studied in reverse mathematics (Simpson 1999).

205.1 Definition

Main article: Ordinal collapsing function

• Ω0 = 0 , and Ωn = ℵn for n > 0.

• Ci(α) is the smallest set of ordinals that contains Ωn for n finite, and contains all ordinals less than Ωi , and isclosed under ordinal addition and exponentiation, and contains Ψj(ξ) if j ≥ i and ξ ∈ Ci(α) and ξ < α .

• Ψi(α) is the smallest ordinal not in Ci(α)

205.2 References• G. Takeuti, Proof theory, 2nd edition 1987 ISBN 0-444-10492-5

• K. Schütte, Proof theory, Springer 1977 ISBN 0-387-07911-4

• Simpson, StephenG. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), CambridgeUniversity Press, ISBN 978-0-521-88439-6, MR 2517689

239

Chapter 206

Ω-logic

Not to be confused with ω-logic.

In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part ofan attempt to generalize the theory of determinacy of pointclasses to cover the structure Hℵ2 . Just as the axiom ofprojective determinacy yields a canonical theory ofHℵ1 , he sought to find axioms that would give a canonical theoryfor the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis isfalse.Woodin’s Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most resultsin the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completenesstheorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive overHℵ2

(inΩ-logic), it must imply that the continuum is not ℵ1 . Woodin also isolated a specific axiom, a variation of Martin’smaximum, which states that any Ω-consistentΠ2 (overHℵ2

) sentence is true; this axiom implies that the continuumis ℵ2 .Woodin also related his Ω-conjecture to a proposed abstract definition of large cardinals: he took a “large cardinalproperty” to be a Σ2 property P (α) of ordinals which implies that α is a strong inaccessible, and which is invariantunder forcing by sets of cardinal less than α. Then the Ω-conjecture implies that if there are arbitrarily large modelscontaining a large cardinal, this fact will be provable in Ω-logic.The theory involves a definition of Ω-validity: A statement is an Ω-valid consequence of a set theory T if it holds inevery model of T having the form V B

α for some ordinal α and some forcing notion B . This notion is clearly preservedunder forcing, and in the presence of a proper class ofWoodin cardinals it will also be invariant under forcing (in otherwords, Ω-satisfiability is preserved under forcing as well). There is also a notion of Ω-provability; here the “proofs”consist of universally Baire sets and are checked by verifying that for every countable transitive model of the theory,and every forcing notion in the model, the generic extension of the model (as calculated in V) contains the “proof”,restricted its own reals. For a proof-setA the condition to be checked here is called "A-closed”. A complexity measurecan be given on the proofs by their ranks in the Wadge hierarchy. Woodin showed that this notion of “provability”implies Ω-validity for sentences which are Π2 over V. The Ω-conjecture states that the converse of this result alsoholds. In all currently known core models, it is known to be true; moreover the consistency strength of the largecardinals corresponds to the least proof-rank required to “prove” the existence of the cardinals.

206.1 References

• Bagaria, Joan; Castells, Neus; Larson, Paul (2006), “An Ω-logic primer”, Set theory (PDF), Trends Math.,Basel, Boston, Berlin: Birkhäuser, pp. 1–28, ISBN 978-3-7643-7691-8, MR 2267144

• Dehornoy, Patrick (2004), “Progrès récents sur l'hypothèse du continu (d'après Woodin)" (PDF), Astérisque(294): 147–172, ISSN 0303-1179, MR 2111643

• Woodin, W. Hugh (1999), The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walterde Gruyter, ISBN 3-11-015708-X, MR 1713438

240

206.2. EXTERNAL LINKS 241

• Woodin,W. Hugh (2001), “The continuum hypothesis. I” (PDF),Notices of the AmericanMathematical Society48 (6): 567–576, ISSN 0002-9920, MR 1834351

• Woodin, W. Hugh (2001b), “The Continuum Hypothesis, Part II” (PDF), Notices of the AMS 48 (7): 681–690

• Woodin, W. Hugh (2005), “The continuum hypothesis”, in Cori, Rene; Razborov, Alexander; Todorcevic,Stevo; et al., Logic Colloquium 2000, Lect. Notes Log. 19, Urbana, IL: Assoc. Symbol. Logic, pp. 143–197,MR 2143878

206.2 External links• W. H. Woodin Slides for 3 talks

242 CHAPTER 206. Ω-LOGIC

206.3 Text and image sources, contributors, and licenses

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RjwilmsiBot, Tijfo098, ForgottenHistory and Anonymous: 1

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• AD+ Source: https://en.wikipedia.org/wiki/AD%2B?oldid=646851881 Contributors: Charles Matthews, Jackol, MFH, Salix alba, Trova-tore, SmackBot, RDBury, JoshuaZ, Mets501, Leyo, Hans Adler, Estudiarme, FrescoBot, Full-date unlinking bot and Anonymous: 1

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244 CHAPTER 206. Ω-LOGIC

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• Difference hierarchy Source: https://en.wikipedia.org/wiki/Difference_hierarchy?oldid=688279888Contributors: Zundark, EmilJ,MarSch,Coremodel, CBM, Pascal.Tesson, David Eppstein, Hans Adler, Yobot, Omnipaedista, Erik9bot and Anonymous: 1

• Double recursion Source: https://en.wikipedia.org/wiki/Double_recursion?oldid=587235152 Contributors: Michael Hardy, Sligocki,CBM, Cydebot, DemocraticLuntz and Anonymous: 1

• Double turnstile Source: https://en.wikipedia.org/wiki/Double_turnstile?oldid=681118137Contributors: Paul A, Hyacinth, GPHemsley,Jason Quinn, Apokrif, DePiep, Dbmag9, Arthur Rubin, Javalenok, Leon..., Gregbard, Cydebot, David Eppstein, Plastikspork, Yobot,Tbvdm, LilHelpa, Xqbot, SporkBot, Paulmiko and Anonymous: 4

• Effective descriptive set theory Source: https://en.wikipedia.org/wiki/Effective_descriptive_set_theory?oldid=643982249 Contribu-tors: Fbkintanar, Salix alba, Trovatore, That Guy, From That Show!, SmackBot, CBM, Hans Adler, Citation bot, George Pelltier, Brirushand Anonymous: 2

• Effective Polish space Source: https://en.wikipedia.org/wiki/Effective_Polish_space?oldid=626071484 Contributors: Tobias Berge-mann, CBM, Cydebot, RebelRobot, CBM2, Andrewbt and Helpful Pixie Bot

• Elementary definition Source: https://en.wikipedia.org/wiki/Elementary_definition?oldid=327373287 Contributors: Michael Hardy,AshtonBenson, Durova, Classicalecon, Hans Adler, Pcap and Erik9bot

• Elementary diagram Source: https://en.wikipedia.org/wiki/Elementary_diagram?oldid=545907035Contributors: Michael Hardy, CBM,Addbot and Luckas-bot

• Elementary sentence Source: https://en.wikipedia.org/wiki/Elementary_sentence?oldid=316831275 Contributors: Michael Hardy andAshtonBenson

• Elementary theory Source: https://en.wikipedia.org/wiki/Elementary_theory?oldid=415833358 Contributors: AshtonBenson and Tku-vho

• End extension Source: https://en.wikipedia.org/wiki/End_extension?oldid=266956794 Contributors: Michael Hardy, Malcolma, Turmsand Ksbrown

• Epsilon-induction Source: https://en.wikipedia.org/wiki/Epsilon-induction?oldid=622266874 Contributors: Michael Hardy, CharlesMatthews, Bennylin, EmilJ, Blotwell, Ruud Koot, MarSch, Salix alba, Petter Strandmark, Mets501, JRSpriggs, CBM, Julian Mendez,Jobu0101, Oddwald, Hans Adler, Erik9bot, Tkuvho and Anonymous: 7

• Equisatisfiability Source: https://en.wikipedia.org/wiki/Equisatisfiability?oldid=550551298Contributors: ObradovicGoran, Tizio, Greg-bard, David Eppstein, AnomieBOT, EmausBot and Anonymous: 4

206.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 245

• Erasure (logic) Source: https://en.wikipedia.org/wiki/Erasure_(logic)?oldid=481083354 Contributors: Graeme Bartlett, Sstrader, Er-icGPrud, SmackBot, Imaginationac, Gregbard, David Eppstein, AlexNewArtBot, Twinsday and Anonymous: 1

• Erdős cardinal Source: https://en.wikipedia.org/wiki/Erd%C5%91s_cardinal?oldid=626044308 Contributors: TakuyaMurata, Schnee-locke, Charles Matthews, Prumpf, Elwoz, Dmytro, Tobias Bergemann, Adam78, Urhixidur, Ben Standeven, Aranel, EmilJ, PaulHanson,Oleg Alexandrov, R.e.b., Trovatore, Bluebot, JRSpriggs, Myasuda, Headbomb, Hans Adler, Yobot, Citation bot, Helpful Pixie Bot, Markviking and Anonymous: 4

• Extender (set theory) Source: https://en.wikipedia.org/wiki/Extender_(set_theory)?oldid=674666272 Contributors: Ben Standeven,Leyo, Hans Adler, Cnwilliams, Chimpionspeak, BG19bot, BattyBot and Mkoeberl

• Extendible cardinal Source: https://en.wikipedia.org/wiki/Extendible_cardinal?oldid=674666291 Contributors: TakuyaMurata, CharlesMatthews, Dmytro, Rgrg, Ben Standeven, Rjwilmsi, R.e.b., Trovatore, SmackBot, Mets501, JRSpriggs, Headbomb, Andrewbt, HansAdler, Citation bot, BattyBot and Anonymous: 2

• Extension (predicate logic) Source: https://en.wikipedia.org/wiki/Extension_(predicate_logic)?oldid=502020968 Contributors: PaulAugust, Salix alba, SmackBot, Nbarth, AndrewWarden, Lambiam, CBM, Gregbard, Ktr101, RichardBergmair, Erik9bot, SD5bot andAK456

• Extensionality Source: https://en.wikipedia.org/wiki/Extensionality?oldid=640783551 Contributors: Charles Matthews, Hyacinth, Pop-ulus, Tobias Bergemann, Snobot, Caesura, Oleg Alexandrov, Linas, BD2412, Seliopou, Hairy Dude, SmackBot, Allixpeeke, Mhss, Blue-bot, Byelf2007, Pezant, CBM, Gregbard, David Eppstein, AlleborgoBot, OKBot, Classicalecon, Mild Bill Hiccup, Addbot, TaBOT-zerem, AnomieBOT, Samppi111, The Wiki ghost, D'ohBot, Andrew Cave, D.Lazard, Ihaveacatonmydesk and Anonymous: 10

• Fiber (mathematics) Source: https://en.wikipedia.org/wiki/Fiber_(mathematics)?oldid=692247679Contributors: Chinju, CharlesMatthews,Oleg Alexandrov, Christopher Thomas, MarSch, LkNsngth, Jon Awbrey, Krasnoludek, JRSpriggs, CBM,Kilva, OrenBochman, Camrn86,LokiClock, Dmcq, JP.Martin-Flatin, Addbot, Ptbotgourou, Ciphers, Erik9bot, Artem M. Pelenitsyn, ZéroBot, Beaumont877, Qetuth,SillyBunnies and Anonymous: 9

• Finite character Source: https://en.wikipedia.org/wiki/Finite_character?oldid=544108057Contributors: Michael Hardy, CharlesMatthews,Aleph4, Giftlite, Paul August, Salix alba, YurikBot, Arthur Rubin, Judicael, Dreadstar, David Eppstein, KittyHawker, Addbot, Ptbot-gourou, 777sms and Anonymous: 1

• Friedberg numbering Source: https://en.wikipedia.org/wiki/Friedberg_numbering?oldid=526274966Contributors: Michael Hardy, CBMand Kumioko

• Gabbay’s separation theorem Source: https://en.wikipedia.org/wiki/Gabbay’s_separation_theorem?oldid=675446270 Contributors:Michael Hardy, BlueNovember, SmackBot, Gregbard, David Eppstein, R'n'B, KCinDC, Yobot, DrilBot and Solomon7968

• Game-theoretic rough sets Source: https://en.wikipedia.org/wiki/Game-theoretic_rough_sets?oldid=626374744 Contributors: MichaelHardy, Malcolma, Magioladitis, David Eppstein, Yobot, AnomieBOT, John of Reading, BattyBot, OccultZone, Nouman.azam1982 andAnonymous: 2

• Generic filter Source: https://en.wikipedia.org/wiki/Generic_filter?oldid=607158000 Contributors: Michael Hardy, Charles Matthews,Tobias Bergemann, Trovatore, Baarslag, Cydebot, Hans Adler, MystBot, Addbot, Yobot, Chimpionspeak and Anonymous: 1

• Ground axiom Source: https://en.wikipedia.org/wiki/Ground_axiom?oldid=610751983Contributors: Rjwilmsi, R.e.b., Headbomb, DavidEppstein, Jesse V., BattyBot, DoctorKubla, Jochen Burghardt and Nynj

• Hartogs number Source: https://en.wikipedia.org/wiki/Hartogs_number?oldid=671323333 Contributors: Zundark, Chinju, CharlesMatthews, Jitse Niesen, Merovingian, Giftlite, Freakofnurture, Trovatore, Arthur Rubin, SmackBot, FlashSheridan, Ligulembot, Ja-son22~enwiki, JRSpriggs, Thijs!bot, Bwhack, Kope, Daniele.tampieri, Michel421, Hans Adler, Addbot, DOI bot, Lightbot, Luckas-bot,Amirobot, Citation bot, MauritsBot, Xqbot, Citation bot 1, JamesMazur22, FilipMaric, DerSpezialist, BendelacBOT, ChrisGualtieri,Monkbot and Anonymous: 8

• Herbrand interpretation Source: https://en.wikipedia.org/wiki/Herbrand_interpretation?oldid=687935553Contributors: CharlesMatthews,CALR, Linas, Zero sharp, CBM, Gregbard, JaGa, Cyborg1, Hans Adler, Addbot, Proofreader77, Waheedghumman, Omnipaedista,Blas3nik and Anonymous: 3

• Hereditarily countable set Source: https://en.wikipedia.org/wiki/Hereditarily_countable_set?oldid=622745897 Contributors: TobiasBergemann, Mhss, Mets501, Rschwieb, JRSpriggs, Hans Adler, Addbot, Unzerlegbarkeit, Dvtausk and Anonymous: 1

• Hereditary set Source: https://en.wikipedia.org/wiki/Hereditary_set?oldid=635237397 Contributors: The Anome, EmilJ, Stemonitis,Salix alba, Trovatore, Ligulembot, JRSpriggs, CBM, Natsirtguy, Squids and Chips, Decoratrix, Watchduck, Hans Adler, 1ForTheMoney,Citation bot and Brirush

• Heyting arithmetic Source: https://en.wikipedia.org/wiki/Heyting_arithmetic?oldid=694508853 Contributors: Chinju, Markhurd, Hy-acinth, Waltpohl, BD2412, SmackBot, CBM, Sdorrance, Gregbard, Cydebot, Magioladitis, R'n'B, Radagast3, Nnemo, El bot de la dieta,Addbot, Yobot, 9258fahsflkh917fas, LilHelpa, ZéroBot, Brirush and Anonymous: 7

• High (computability) Source: https://en.wikipedia.org/wiki/High_(computability)?oldid=681502162Contributors: Tenebrae,Wknight94,SmackBot, CBM, Cydebot, Althai, Quux0r, Birdstends, MarcelB612 and Anonymous: 1

• Hilbert–Bernays provability conditions Source: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions?oldid=680546770 Contributors: Tobias Bergemann, EmilJ, Ruud Koot, CBM, Yobot, Tkuvho and Anonymous: 1

• Homogeneous (large cardinal property) Source: https://en.wikipedia.org/wiki/Homogeneous_(large_cardinal_property)?oldid=449295994Contributors: JRSpriggs, CBM, Hans Adler, Erik9bot and BrideOfKripkenstein

• Homogeneous tree Source: https://en.wikipedia.org/wiki/Homogeneous_tree?oldid=605828305 Contributors: Mrwright, Hans Adler,DOI bot, Citation bot 1 and BattyBot

• Homogeneously Suslin set Source: https://en.wikipedia.org/wiki/Homogeneously_Suslin_set?oldid=605828312 Contributors: Dman-ning, Mrwright, SmackBot, McPoet, Zrustin, Hans Adler, DOI bot, Citation bot 1 and BattyBot

• Honest leftmost branch Source: https://en.wikipedia.org/wiki/Honest_leftmost_branch?oldid=676093878Contributors: Zundark,MichaelHardy, TakuyaMurata, Andreas Kaufmann, MFH, Algebraist, David Eppstein, Cobi, COBot and Solomon7968

246 CHAPTER 206. Ω-LOGIC

• Indiscernibles Source: https://en.wikipedia.org/wiki/Indiscernibles?oldid=635482675 Contributors: Karada, Charles Matthews, BenStandeven, Adamarthurryan, Dfass, Mets501, Zero sharp, JRSpriggs, CBM, Hans Adler, Erik9bot, Deltahedron and Brirush

• Inductive set Source: https://en.wikipedia.org/wiki/Inductive_set?oldid=607144947Contributors: TakuyaMurata, Trovatore, SmackBot,PieRRoMaN, Ligulembot, Hans Adler, Yobot, Citation bot, Kephir and Anonymous: 3

• Ineffable cardinal Source: https://en.wikipedia.org/wiki/Ineffable_cardinal?oldid=543711107 Contributors: Vicki Rosenzweig, Takuya-Murata, Schneelocke, Charles Matthews, Aleph4, Tobias Bergemann, Eoghan, Ben Standeven, Oleg Alexandrov, Linas, R.e.b., GoOd-CoNtEnT, Mets501, JRSpriggs, Norman314, David Eppstein, LokiClock, Hans Adler, Addbot, Citation bot 1, ZéroBot and Anonymous:5

• Infinite descending chain Source: https://en.wikipedia.org/wiki/Infinite_descending_chain?oldid=682612200 Contributors: Andre En-gels, Patrick, Michael Hardy, Wshun, Charles Matthews, MatrixFrog, Markus Krötzsch, Quarl, Rich Farmbrough, Salix alba, NavarroJ,SmackBot, Mhss, Nbarth, CBM, David Eppstein, Erik9bot, Helpful Pixie Bot, KLBot2, Solomon7968, Ansatz and Anonymous: 8

• Information diagram Source: https://en.wikipedia.org/wiki/Information_diagram?oldid=662925968Contributors: PAR,Mendaliv, Deep-math, Samadony, KonradVoelkel, Qetuth, Unitknow2, JaconaFrere and Anonymous: 6

• Institutional model theory Source: https://en.wikipedia.org/wiki/Institutional_model_theory?oldid=626256433 Contributors: Mdd,Tillmo, Arthur Rubin, SmackBot, Lambiam, CBM, Kyriosity, Tcamps42, Hans Adler, Jochen Burghardt and Anonymous: 1

• Iterable cardinal Source: https://en.wikipedia.org/wiki/Iterable_cardinal?oldid=678241895 Contributors: Michael Hardy, R.e.b., Mel-choir and K9re11

• Jensen’s covering theorem Source: https://en.wikipedia.org/wiki/Jensen’s_covering_theorem?oldid=451291546Contributors: MichaelHardy, R.e.b., CBM, Kope, Citation bot, Xqbot, Omnipaedista, RjwilmsiBot and CitationCleanerBot

• Joint embedding property Source: https://en.wikipedia.org/wiki/Joint_embedding_property?oldid=490338555 Contributors: MichaelHardy, Kuratowski’s Ghost, Bender2k14, Hans Adler, Xx521xx and Anonymous: 1

• Judgment (mathematical logic) Source: https://en.wikipedia.org/wiki/Judgment_(mathematical_logic)?oldid=600394251 Contribu-tors: Michael Hardy, BD2412, Physis, Gregbard, Deadbeef, GoingBatty, JPaestpreornJeolhlna, PlaidPolarity and Anonymous: 2

• Jónsson cardinal Source: https://en.wikipedia.org/wiki/J%C3%B3nsson_cardinal?oldid=655323891 Contributors: TakuyaMurata, BenStandeven, EmilJ, R.e.b., Trovatore, Luminus1, JRSpriggs, CBM, Kope, Hans Adler, Yobot, Citation bot, Trappist the monk, RjwilmsiBotand Anonymous: 1

• Kanamori–McAloon theorem Source: https://en.wikipedia.org/wiki/Kanamori%E2%80%93McAloon_theorem?oldid=569642732Con-tributors: Michael Hardy, Giftlite, Rjwilmsi, R.e.b., Sodin, CBM, Headbomb, David Eppstein, Yobot and Anonymous: 1

• Kleene–Rosser paradox Source: https://en.wikipedia.org/wiki/Kleene%E2%80%93Rosser_paradox?oldid=628611729 Contributors:William Avery, Michael Hardy, Hyacinth, Leonard G., Beland, Rich Farmbrough, Bender235, Mike Schwartz, Spug, Linas, Smack-Bot, CapitalSasha, Lambiam, Zero sharp, CBM, Falcor84, Daniel5Ko, Dorftrottel, Seanmclaughlin, Paradoctor, Neuralwarp, Marc vanLeeuwen, C. A. Russell, Yobot, Pcap, DmitriyZotikov, Jochen Burghardt and Anonymous: 12

• Knaster’s condition Source: https://en.wikipedia.org/wiki/Knaster’s_condition?oldid=632652335Contributors: Michael Hardy,Woohookitty,Myasuda, David Eppstein, SchreiberBike, Yaddie, Helpful Pixie Bot, Brad7777 and Mark viking

• Kunen’s inconsistency theorem Source: https://en.wikipedia.org/wiki/Kunen’s_inconsistency_theorem?oldid=693735606 Contribu-tors: Michael Hardy, Dmytro, Rjwilmsi, R.e.b., JRSpriggs, CBM, N5iln, Magioladitis, David Eppstein, Hans Adler, Yobot, Citation bot,Trappist the monk and Anonymous: 2

• Kuratowski’s free set theorem Source: https://en.wikipedia.org/wiki/Kuratowski’s_free_set_theorem?oldid=648947270 Contribu-tors: Aecis, Rgdboer, Kusma, Oleg Alexandrov, Salix alba, Malcolma, Egpetersen, David Eppstein, Kope, R'n'B, Fwehrung, Hans Adler,Yobot, K9re11 and Anonymous: 4

• Kurepa tree Source: https://en.wikipedia.org/wiki/Kurepa_tree?oldid=668596608 Contributors: Michael Hardy, Waltpohl, AndreasKaufmann, Woohookitty, R.e.b., CmdrObot, CBM, Picaroon, Tokenzero, David Eppstein, Epsilon0, Kope, Dlinetsky, Hans Adler, Myst-Bot, Addbot, KamikazeBot, Citation bot, FrescoBot, Daysrr, Deltahedron and Anonymous: 9

• Language equation Source: https://en.wikipedia.org/wiki/Language_equation?oldid=674437439Contributors: Zzyzx11,Wimt, ThomasConnor, Hermel, Hyperyl, Okhotin, Qetuth, Jochen Burghardt and Anonymous: 1

• Laver function Source: https://en.wikipedia.org/wiki/Laver_function?oldid=648980175 Contributors: Michael Hardy, Discospinster,Rjwilmsi, Bgwhite, SmackBot, CBM, Kope, Flyer22 Reborn, Hans Adler, Deltahedron, K9re11 and Anonymous: 1

• Least fixed point Source: https://en.wikipedia.org/wiki/Least_fixed_point?oldid=699726985 Contributors: Michael Hardy, CharlesMatthews, Dcoetzee, Creidieki, Oleg Alexandrov, Jpbowen, Ott2, Mhss, CRGreathouse, Cydebot, Blaisorblade, Heyitspeter, AaronRotenberg, Addbot, Arthur MILCHIOR, SethFogarty, Tijfo098, Wcherowi, Jochen Burghardt and Anonymous: 8

• LEGO (proof assistant) Source: https://en.wikipedia.org/wiki/LEGO_(proof_assistant)?oldid=546222014 Contributors: Ruud Koot,Cydebot, David Eppstein and Addbot

• Lightface analytic game Source: https://en.wikipedia.org/wiki/Lightface_analytic_game?oldid=631352163 Contributors: Sgeo, Trova-tore, TechnoGuyRob, CBM, AnomieBOT, Erik9bot and Mark viking

• Limitation of size Source: https://en.wikipedia.org/wiki/Limitation_of_size?oldid=635377439 Contributors: Charles Matthews, RyanReich, Salix alba, Trovatore, Arundhati bakshi, SmackBot, Steve Byrne, JRSpriggs, Hans Adler, 1ForTheMoney, Citation bot and Brirush

• Limited principle of omniscience Source: https://en.wikipedia.org/wiki/Limited_principle_of_omniscience?oldid=523663730 Con-tributors: Michael Hardy, CBM and Anonymous: 1

• Lindström’s theorem Source: https://en.wikipedia.org/wiki/Lindstr%C3%B6m’s_theorem?oldid=612860185 Contributors: MichaelHardy, Dcoetzee, Giftlite, Chalst, Nortexoid, Gene Nygaard, Sebkha, Tillmo, Alynna Kasmira, SmackBot, RDBury, Turms, Zero sharp,CBM, Gregbard, MarshBot, David Eppstein, VanishedUserABC, Hans Adler, Addbot, Yobot, Pcap, DrilBot, ZéroBot, Tijfo098, HelpfulPixie Bot, BG19bot and Anonymous: 5

• Linked set Source: https://en.wikipedia.org/wiki/Linked_set?oldid=604894240 Contributors: Michael Hardy, David Eppstein, Yaddie,Helpful Pixie Bot, Brad7777 and Anonymous: 1

206.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 247

• LOGCFL Source: https://en.wikipedia.org/wiki/LOGCFL?oldid=607685060 Contributors: Tizio, Sbrools, SmackBot, Gelingvistoj,David Eppstein, JaGa, R'n'B, Jamelan, Addbot, Yobot, Twri, LucienBOT, RobinK and Anonymous: 1

• Logic for Computable Functions Source: https://en.wikipedia.org/wiki/Logic_for_Computable_Functions?oldid=602596773 Contrib-utors: Mav, Michael Hardy, Michaeln, Greenrd, Tobias Bergemann, Urhixidur, Leibniz, Ascánder, Chalst, Cwolfsheep, Daira Hopwood,Ruud Koot, Hairy Dude, Jpbowen, Gregbard, Cydebot, The Wild Falcon, Legobot, Pcap, Masssly and Anonymous: 3

• Logical assertion Source: https://en.wikipedia.org/wiki/Logical_assertion?oldid=681139081 Contributors: Docu, Silverfish, Dysprosia,Hyacinth, Jag123, DiegoMoya, Oleg Alexandrov, SeventyThree, SAE1962, BOT-Superzerocool, Mhss, Nbarth, Mets501, Peter1c, CBM,Addbot, Nate Wessel, ChuispastonBot, Masssly, Dwellee and Anonymous: 2

• Logical graph Source: https://en.wikipedia.org/wiki/Logical_graph?oldid=602607355 Contributors: Michael Hardy, Goethean, Giftlite,Pmanderson, Paul August, El C, Jeffrey O. Gustafson, Linas, DoubleBlue, Mathbot, RussBot, Trovatore, Closedmouth, C.Fred, Trebor,E946, Jon Awbrey, Mostlyharmless, FlyHigh, Wvbailey, JzG, Dbtfz, Antonielly, Slakr, Mets501, OS2Warp, CRGreathouse, CBM, GogoDodo, Hut 8.5, David Eppstein, Brigit Zilwaukee, Yolanda Zilwaukee, On This Continent, CardinalDan, The Tetrast, Seb26, Coffee,TFCforever, Islaammaged126, Hans Adler, Buchanan’s Navy Sec, Overstay, Marsboat, Unco Guid, Viva La Information Revolution!,Dave Chaparral, Poke Salat Annie, Flower Mound Belle, Navy Pierre, Mrs. Lovett’s Meat Puppets, Chester County Dude, SoutheastPenna Poppa, Delaware Valley Girl, Addbot, Ichiboku Natabori, MrOllie, Queen of the Dishpan, Twri, Throw it in the Fire, C.W.Murry, Branzillo, Lantern Leatherhead, Gamewizard71, Igotta Lemma, Pangur Ban My Cat, A Thousand Dancing Hamsters, Massslyand Anonymous: 7

• Logicalmachine Source: https://en.wikipedia.org/wiki/Logical_machine?oldid=602607667Contributors: Rich Farmbrough, OlegAlexan-drov, Formativ, Bwpach, CBM, Alaibot, Emeraude, The Tetrast, Addbot, Eumolpo, PigFlu Oink, Masssly and Anonymous: 1

• Low (computability) Source: https://en.wikipedia.org/wiki/Low_(computability)?oldid=657249275 Contributors: Johndburger, Smack-Bot, Toughpigs, Cydebot, Althai, Quux0r, MarcelB612, MaximalIdeal, Deltahedron and Anonymous: 1

• Lowbasis theorem Source: https://en.wikipedia.org/wiki/Low_basis_theorem?oldid=665829033Contributors: Tobias Bergemann, OlegAlexandrov, Rjwilmsi, CBM, Cydebot, Althai, David Eppstein, DavidCBryant, Yobot, AnomieBOT, Deltahedron and Anonymous: 2

• Lusin’s separation theorem Source: https://en.wikipedia.org/wiki/Lusin’s_separation_theorem?oldid=616041706Contributors: MichaelHardy, Giftlite, R.e.b., Sodin, CBM, Magioladitis, Yoni, Daniele.tampieri, Helpful Pixie Bot and Anonymous: 1

• Lévy hierarchy Source: https://en.wikipedia.org/wiki/L%C3%A9vy_hierarchy?oldid=657387428Contributors: Michael Hardy, Bearcat,Tsirel, Rjwilmsi, R.e.b., JRSpriggs, Snalin, Magioladitis, Minimiscience, Biscuittin, SchreiberBike, Yobot, Xqbot, Tijfo098, Catrincm,Deltahedron, Mark viking and Anonymous: 1

• Martin’smaximum Source: https://en.wikipedia.org/wiki/Martin’s_maximum?oldid=689864001Contributors: Michael Hardy, CharlesMatthews, Pol098, Rjwilmsi, R.e.b., CBM, Cydebot, Kope, Hans Adler, Citation bot, Citation bot 1, DrilBot, Deltahedron and Anony-mous: 1

• Material nonimplication Source: https://en.wikipedia.org/wiki/Material_nonimplication?oldid=694579665Contributors: Kaldari, BD2412,Kbdank71, Chris Capoccia, MacMog, SmackBot, Cybercobra, Bjankuloski06en~enwiki, Gregbard, Cydebot, David Eppstein, MauriceCarbonaro, Anzurio, Francvs, Classicalecon, BANZ111, Alex836, Watchduck, Addbot, Meisam, Luckas-bot, Yobot, FrescoBot, JesseV., EmausBot, Jontturi, Matthew Kastor and Anonymous: 6

• Maximal set Source: https://en.wikipedia.org/wiki/Maximal_set?oldid=638561052Contributors: Michael Hardy, Paul August, C S, Con-scious, SmackBot, Cronholm144, CRGreathouse, Cydebot, Alaibot, Jay1279, Fabrictramp, David Eppstein, Frank Stephan, JackSchmidt,JL-Bot, NClement, Citation bot, Citation bot 1, Trappist the monk, Tagib, Deltahedron, PlaidPolarity and Anonymous: 1

• Michael D. Morley Source: https://en.wikipedia.org/wiki/Michael_D._Morley?oldid=659227815 Contributors: Michael Hardy, Aleph4,Gene Nygaard, Alai, GregAsche, Jaxl, Caerwine, Bluebot, Mr. Lefty, AlsatianRain, Cydebot, Waacstats, Johnpacklambert, Feepflop,Hans Adler, Addbot, Yobot, Laubbaum, Omnipaedista, Joeylinpc, Suslindisambiguator and Anonymous: 3

• Milner–Rado paradox Source: https://en.wikipedia.org/wiki/Milner%E2%80%93Rado_paradox?oldid=618654006Contributors: MichaelHardy, R.e.b., CBM, Gregbard, Headbomb, David Eppstein, Kope, Allispaul, Yobot, Vilimlendvaj and Anonymous: 2

• Minimal logic Source: https://en.wikipedia.org/wiki/Minimal_logic?oldid=624352098 Contributors: Markhurd, EmilJ, Clconway, Cy-debot, A3nm, Addbot, DixonDBot, Hpvpp, Chricho, Jochen Burghardt and Anonymous: 1

• Nice name Source: https://en.wikipedia.org/wiki/Nice_name?oldid=691868391 Contributors: Michael Hardy, Rich Farmbrough, C S,Bhny, Arthur Rubin, Radagast83, Nagle, Jason22~enwiki, JRSpriggs, CBM,Cydebot, Alphachimpbot, Tomaxer, HansAdler, AnomieBOT,Erik9bot, DARTH SIDIOUS 2, ClueBot NG, Helpful Pixie Bot, Ansatz, Pastisch and Anonymous: 3

• Normalmeasure Source: https://en.wikipedia.org/wiki/Normal_measure?oldid=607169374Contributors: CharlesMatthews, JRSpriggs,CBM, Hans Adler, Yobot and Erik9bot

• Omega-categorical theory Source: https://en.wikipedia.org/wiki/Omega-categorical_theory?oldid=538098728 Contributors: MichaelHardy, R.e.b., RDBury, CBM, VanishedUserABC, Hans Adler, False vacuum, Ultracoffee, BG19bot, Deltahedron and Anonymous: 1

• Ordinal definable set Source: https://en.wikipedia.org/wiki/Ordinal_definable_set?oldid=627021132 Contributors: Zundark, MichaelHardy, Aleph4, Ben Standeven, R.e.b., Trovatore, Arthur Rubin, SmackBot, Lambiam, David Eppstein, Pepve, Andrewbt, Hans Adler,Addbot, Luckas-bot, BrideOfKripkenstein, Trappist the monk, Helpful Pixie Bot, ChrisGualtieri and Anonymous: 2

• Ordinal logic Source: https://en.wikipedia.org/wiki/Ordinal_logic?oldid=490075488 Contributors: Michael Hardy, JRSpriggs, Greg-bard, VanishedUserABC, Tinton5, Helpful Pixie Bot, Brad7777 and Qetuth

• Paraconsistentmathematics Source: https://en.wikipedia.org/wiki/Paraconsistent_mathematics?oldid=602434485Contributors: JohnOwens,Charles Matthews, Henrygb, Eduardoporcher, Mcsee, Porcher, Joelr31, SmackBot, CBM, Gregbard, David Eppstein, Philg88, Embra-ceParadox, Aubreybardo and Anonymous: 3

• Polyadic algebra Source: https://en.wikipedia.org/wiki/Polyadic_algebra?oldid=572263580Contributors: Michael Hardy, Giftlite, Ezhiki,Mdd, Rjwilmsi, Pthag, CBM, Iohannes Animosus, Hugo Herbelin, Charvest, Tijfo098, Helpful Pixie Bot, Jochen Burghardt and Anony-mous: 1

248 CHAPTER 206. Ω-LOGIC

• Predicate logic Source: https://en.wikipedia.org/wiki/Predicate_logic?oldid=699122914 Contributors: Toby Bartels, Michael Hardy,Andres, Hyacinth, Robbot, MathMartin, Giftlite, Leonard G., Mindmatrix, Thekohser, Eubot, Chobot, Sharkface217, Jpbowen, Tomisti,SmackBot, Mhss, Cybercobra, Nakon, Byelf2007, Wvbailey, Bjankuloski06en~enwiki, George100, CBM, Gregbard, Naudefj, Thijs!bot,EdJohnston, JAnDbot, Hypergeek14, Stassa, Vanished user g454XxNpUVWvxzlr, Policron, Dessources, JohnBlackburne, AnonymousDissident, Gerakibot, Soler97, Kumioko, DesolateReality, Xiaq, ClueBot, Taxa, Djk3, TimClicks, Addbot, Jayde239, Yobot, AnomieBOT,Materialscientist, RandomDSdevel, ESSch, Keri, Logichulk, Xnn, EmausBot, WikitanvirBot, Mayur, ClueBot NG, Satellizer, ChesterMarkel, MerlIwBot, Helpful Pixie Bot, Virago250, Brad7777, Jochen Burghardt, BoltonSM3, Tomajohnson and Anonymous: 35

• Principle of distributivity Source: https://en.wikipedia.org/wiki/Principle_of_distributivity?oldid=582122861 Contributors: CharlesMatthews, Tobias Bergemann, Chalst, Oleg Alexandrov, Btyner, Rjwilmsi, Jb-adder, Shirahadasha, CBM, Gregbard, Cydebot, Erudecorp,Gzhanstong, LiederLover1982, AnomieBOT, Erik9bot, Gamewizard71, Eskilp, Mark viking and Anonymous: 6

• Projection (set theory) Source: https://en.wikipedia.org/wiki/Projection_(set_theory)?oldid=675920215 Contributors: Michael Hardy,Charles Matthews, Oleg Alexandrov, SmackBot, Nbarth, Jon Awbrey, Lambiam, CBM, David Eppstein, Erik9bot, No One of Conse-quence, Zfeinst, Solomon7968, JMP EAX and Anonymous: 4

• Proof compression Source: https://en.wikipedia.org/wiki/Proof_compression?oldid=683617033Contributors: Michael Hardy,Mr. Stradi-varius, Ceilican, LilHelpa, BG19bot, ChrisGualtieri, Mark viking, Ezequiel234 and Anonymous: 1

• Proofmining Source: https://en.wikipedia.org/wiki/Proof_mining?oldid=442037556Contributors: Michael Hardy, Jayme, CRGreathouse,CBM, Classicalecon, Pauloboliva, Unzerlegbarkeit, Yobot, Willy xD and Anonymous: 1

• Pseudo-intersection Source: https://en.wikipedia.org/wiki/Pseudo-intersection?oldid=648937330 Contributors: R.e.b. and K9re11• Pseudo-order Source: https://en.wikipedia.org/wiki/Pseudo-order?oldid=537194190 Contributors: Toby Bartels, Cydebot, David Epp-

stein, Unzerlegbarkeit, Yobot, DrilBot, EefeG0hi and Anonymous: 1• Ramified forcing Source: https://en.wikipedia.org/wiki/Ramified_forcing?oldid=476916548Contributors: Michael Hardy, Tobias Berge-

mann, Rjwilmsi, R.e.b., Trovatore, CBM, Cydebot, David Eppstein, Mild Bill Hiccup, Hans Adler, Citation bot, Citation bot 1 andAnonymous: 1

• Ramsey cardinal Source: https://en.wikipedia.org/wiki/Ramsey_cardinal?oldid=644349375Contributors: Schneelocke, CharlesMatthews,Dmytro, Aleph4, Tobias Bergemann, Ben Standeven, EmilJ, R.e.b., Vclaw, Trovatore, SmackBot, JRSpriggs, Myasuda, S Marshall, Aer-vanath, SieBot, Hans Adler, Addbot, Citation bot, Anne Bauval, Trappist the monk, ZéroBot, Mirror symmetry and Anonymous: 1

• Rank-into-rank Source: https://en.wikipedia.org/wiki/Rank-into-rank?oldid=677698795Contributors: Zundark,Michael Hardy, Takuya-Murata, Julesd, Schneelocke, Charles Matthews, Jitse Niesen, Dmytro, GeneWard Smith, Oleg Alexandrov, Josh Parris, Rjwilmsi, R.e.b.,Trovatore, Luminus1, JRSpriggs, CBM, Headbomb, Widefox, Hans Adler, AnomieBOT, Citation bot, Citation bot 1 and Anonymous: 5

• Recursive ordinal Source: https://en.wikipedia.org/wiki/Recursive_ordinal?oldid=482901449 Contributors: Zundark, Falsifian, Spam-bit, Oleg Alexandrov, Salix alba, R.e.b., JRSpriggs, CBM, Cydebot, Hans Adler, FrescoBot, Burritoburritoburrito and Anonymous: 2

• Reduced product Source: https://en.wikipedia.org/wiki/Reduced_product?oldid=481254985 Contributors: Michael Hardy, Waltpohl,EmilJ, MarSch, R.e.b., Malcolma, MrShamrock, SmackBot, Popopp and Helpful Pixie Bot

• Redundant proof Source: https://en.wikipedia.org/wiki/Redundant_proof?oldid=607430635 Contributors: Michael Hardy, Bearcat,Boomur, D.Lazard, Ad Orientem, Mark viking and Ezequiel234

• Reflecting cardinal Source: https://en.wikipedia.org/wiki/Reflecting_cardinal?oldid=637751025 Contributors: Michael Hardy, R.e.b.,CBM, Magioladitis, Kope, BrianS36, Hans Adler, Yobot and Citation bot

• Remarkable cardinal Source: https://en.wikipedia.org/wiki/Remarkable_cardinal?oldid=614569729 Contributors: Schneelocke, TobiasBergemann, Everyking, BenjBot, Oleg Alexandrov, R.e.b., JRSpriggs, Hans Adler, Qwfp, CitationCleanerBot and Anonymous: 4

• Richardson’s theorem Source: https://en.wikipedia.org/wiki/Richardson’s_theorem?oldid=665153030 Contributors: Michael Hardy,Dominus, Giftlite, Jason Quinn, Profzoom, Jason Davies, Mandarax, R.e.b., Jfriedl, Spacepotato, RDBury, CRGreathouse, CBM, Greg-bard, Cydebot, Asmeurer, Laurusnobilis, Haseldon, Gaz v pol, Addbot, AndersBot, AnomieBOT, Citation bot, RedZiz~enwiki, Citationbot 1, Chricho, D.Lazard, Helpful Pixie Bot, Syedhanif86, Kondormari, Cyrapas and Anonymous: 12

• Robinson’s joint consistency theorem Source: https://en.wikipedia.org/wiki/Robinson’s_joint_consistency_theorem?oldid=671837343Contributors: Giftlite, Waltpohl, Tillmo, RDBury, CBM, Gregbard, DavidCBryant, Addbot, Amirobot, Stefan.vatev, Trappist the monkand Helpful Pixie Bot

• Rowbottom cardinal Source: https://en.wikipedia.org/wiki/Rowbottom_cardinal?oldid=627081203Contributors: Ben Standeven, EmilJ,Allen3, R.e.b., Trovatore, SmackBot, Luminus1, Luminus2, JRSpriggs, Hans Adler, Yobot, Citation bot and Trappist the monk

• Scattered order Source: https://en.wikipedia.org/wiki/Scattered_order?oldid=568837076 Contributors: Michael Hardy, Aleph4, EmilJ,Rjwilmsi, CBM, Giggy, David Eppstein, BOTijo and CitationCleanerBot

• Semicomputable function Source: https://en.wikipedia.org/wiki/Semicomputable_function?oldid=465074972 Contributors: MichaelHardy, AtZeuS, Malcolma, SmackBot, CBM, Yobot, Erik9bot and Anonymous: 2

• Separating set Source: https://en.wikipedia.org/wiki/Separating_set?oldid=636544832Contributors: Michael Hardy, JitseNiesen,Math-Martin, Giftlite, Andreas Kaufmann, Salix alba, Silly rabbit, CBM, David Eppstein, Foxj, AnomieBOT, Schmittz and Anonymous: 1

• Set constraint Source: https://en.wikipedia.org/wiki/Set_constraint?oldid=679245960 Contributors: Qwertyus, Rjwilmsi, LogAntiLog,BG19bot, Jochen Burghardt and Monkbot

• Set function Source: https://en.wikipedia.org/wiki/Set_function?oldid=687421102 Contributors: Michael Hardy, Gabbe, Andres, RD-Bury, Kjetil1001, CBM, David Eppstein, Addbot, HRoestBot and Anonymous: 3

• Shelah cardinal Source: https://en.wikipedia.org/wiki/Shelah_cardinal?oldid=457961686 Contributors: Schneelocke, Tobias Berge-mann, Bender235, Ben Standeven, BenjBot, Spambit, Oleg Alexandrov, JRSpriggs, CBM and Hans Adler

• Shrewd cardinal Source: https://en.wikipedia.org/wiki/Shrewd_cardinal?oldid=607167441 Contributors: Ben Standeven, R.e.b., Head-bomb, Hans Adler, Debresser and Yobot

206.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 249

• Soft set Source: https://en.wikipedia.org/wiki/Soft_set?oldid=688511902 Contributors: Michael Hardy, Rjwilmsi, Alcides, SmackBot,YellowMonkey, RayAYang, JustAnotherJoe, Freelance Intellectual, David Eppstein, LokiClock, Kharal9, De728631, Addbot, Yobot,Xqbot, RjwilmsiBot, Matsievsky, StarryGrandma, Cerabot~enwiki, Atharkharal, Monkbot, Rknmohanty and Anonymous: 2

• Square principle Source: https://en.wikipedia.org/wiki/Square_principle?oldid=678712472 Contributors: Charles Matthews, Barnabydawson, Oleg Alexandrov, R.e.b., CBM, David Eppstein, Hans Adler, Yobot, Erik9bot, Paolo Lipparini and Anonymous: 4

• Strength (mathematical logic) Source: https://en.wikipedia.org/wiki/Strength_(mathematical_logic)?oldid=545998336 Contributors:Gregbard, VanishedUserABC, Addbot, The Interior, Helpful Pixie Bot and Anonymous: 3

• Strong cardinal Source: https://en.wikipedia.org/wiki/Strong_cardinal?oldid=618575058 Contributors: TakuyaMurata, Schneelocke,Charles Matthews, Dmytro, Aleph4, Blainster, Tobias Bergemann, Michael Devore, Ben Standeven, ABCD, SmackBot, JRSpriggs, Zo-oloo, Sam Coskey, Hans Adler, Yobot and Citation bot

• Strong partition cardinal Source: https://en.wikipedia.org/wiki/Strong_partition_cardinal?oldid=548262903 Contributors: SimonP,Oleg Alexandrov, Rjwilmsi, Pernishus, R.e.b., CBM, David Eppstein, Hans Adler, Citation bot, Citation bot 1, DrilBot and Chimpi-onspeak

• Strongly compact cardinal Source: https://en.wikipedia.org/wiki/Strongly_compact_cardinal?oldid=607170771 Contributors: Dmytro,Tobias Bergemann, Ben Standeven, SmackBot, Mets501, CBM, Hans Adler, Yobot, Citation bot and Anonymous: 4

• Subcompact cardinal Source: https://en.wikipedia.org/wiki/Subcompact_cardinal?oldid=358407435Contributors: Dmytro, Cmdrjame-son, Mets501, JRSpriggs, CBM, Ntsimp, Auntof6 and Hans Adler

• Subcountability Source: https://en.wikipedia.org/wiki/Subcountability?oldid=582355341Contributors: CBM,Cydebot, Avytipat, R'n'B,Whatever511723421343, Yobot, LudovicoVan and Anonymous: 1

• Subtle cardinal Source: https://en.wikipedia.org/wiki/Subtle_cardinal?oldid=627085014Contributors: Zundark, TakuyaMurata, Schnee-locke, Dmytro, Aleph4, Tobias Bergemann, Ben Standeven, Oleg Alexandrov, R.e.b., Mets501, Hans Adler, Yobot, Citation bot, Citationbot 1, Trappist the monk and Anonymous: 1

• Successor function Source: https://en.wikipedia.org/wiki/Successor_function?oldid=688757936 Contributors: Michael Hardy, Dori,Angela, Charles Matthews, Kaal, Dissident, Pearle, Diego Moya, Oleg Alexandrov, Ruud Koot, Triddle, BD2412, Salix alba, RussBot,Bigmantonyd, CmdrObot, CBM, Nick Number, David Eppstein, Tbvdm, VladikVP, Jochen Burghardt, Sylphi and Anonymous: 6

• Sudan function Source: https://en.wikipedia.org/wiki/Sudan_function?oldid=655501814 Contributors: Charles Matthews, Jerzy, Rameinstein, Factitious, Ben Standeven, Oleg Alexandrov, Hgkamath, Jwy, Ryszard Szopa~enwiki, Thijs!bot, Dfrg.msc, David Eppstein,Epsilon0, Addbot, Lightbot, Yobot, FrescoBot, CarrieVS and Anonymous: 7

• Supernatural number Source: https://en.wikipedia.org/wiki/Supernatural_number?oldid=699263593 Contributors: Michael Hardy,TakuyaMurata, Jason Quinn, Algebraist, Malcolma, CRGreathouse, CBM, Alaibot, JamesBWatson, Jéské Couriano, David Eppstein,Newbyguesses, DFRussia, Addbot, Luckas-bot, Omnipaedista, Aaqqqq, John of Reading, Minimalrho, Deltahedron, Spectral sequence,Tall human, GeoffreyT2000, ScrapIronIV and Anonymous: 5

• Superposition calculus Source: https://en.wikipedia.org/wiki/Superposition_calculus?oldid=611708869 Contributors: Michael Hardy,Charles Matthews, Stephan Schulz, Tizio, Ojcit, CRGreathouse, CBM, Benea, Addbot, Luckas-bot, Wolfgang42, RichardMills65, Markviking and Anonymous: 5

• Superstrong cardinal Source: https://en.wikipedia.org/wiki/Superstrong_cardinal?oldid=674788899Contributors: TakuyaMurata, Schnee-locke, Aleph4, Ben Standeven, Oleg Alexandrov, Trovatore, SmackBot, JRSpriggs, MetsBot, Hans Adler, Citation bot and BattyBot

• Suslin cardinal Source: https://en.wikipedia.org/wiki/Suslin_cardinal?oldid=442018465 Contributors: MFH, Ani td, Hans Adler, Yobotand Chimpionspeak

• Suslin representation Source: https://en.wikipedia.org/wiki/Suslin_representation?oldid=654522616 Contributors: Michael Hardy, An-dreas Kaufmann, Rich Farmbrough, MFH, R.e.b., Trovatore, JRSpriggs, Hans Adler, Tonyxty, RockMagnetist and Anonymous: 1

• Suslin tree Source: https://en.wikipedia.org/wiki/Suslin_tree?oldid=621501329 Contributors: Michael Hardy, Giftlite, Andreas Kauf-mann, R.e.b., CBM, Epsilon0, Cobi, Citation bot, Citation bot 1, Deltahedron, TheMachine999 and Anonymous: 3

• Switching circuit theory Source: https://en.wikipedia.org/wiki/Switching_circuit_theory?oldid=691143277 Contributors: Rpyle731,AJim, Wtshymanski, David Eppstein, Jim.henderson, AnomieBOT, K6ka, Tystnaden, Satellizer, DBigXray, BG19bot, Qetuth, Ramsgr83and Anonymous: 8

• Symmetric set Source: https://en.wikipedia.org/wiki/Symmetric_set?oldid=590021907Contributors: TakuyaMurata, Giftlite, Malcolma,SmackBot, Minimiscience, 777sms, Dfqppp, Qetuth and Mgkrupa

• Systems of Logic Based on Ordinals Source: https://en.wikipedia.org/wiki/Systems_of_Logic_Based_on_Ordinals?oldid=678461600Contributors: Michael Hardy, Aymanshamma, Bejnar, JRSpriggs, Gregbard, Olsonist, Daniel5Ko, Duncan.Hull, VanishedUserABC,Hairhorn, Helpful Pixie Bot, Brad7777, Qetuth, RudolfRed, Dexbot, Charleswfox and Anonymous: 1

• Tail sequence Source: https://en.wikipedia.org/wiki/Tail_sequence?oldid=624335850Contributors: SmackBot, Cronholm144, Jason22~enwiki,Ouedbirdwatcher, Hans Adler and Mark viking

• Takeuti’s conjecture Source: https://en.wikipedia.org/wiki/Takeuti’s_conjecture?oldid=696686530 Contributors: Michael Hardy,TakuyaMurata, CharlesMatthews, Chalst, Mairi, R.e.b., CambridgeBayWeather, ArglebargleIV,Gregbard, David Eppstein, Cobi, Tradereddy,AlptaBot, Anne Bauval, Omnipaedista, FrescoBot, Proof Theorist and Anonymous: 3

• Tarski–Kuratowski algorithm Source: https://en.wikipedia.org/wiki/Tarski%E2%80%93Kuratowski_algorithm?oldid=551279029Con-tributors: Psychonaut, MathMartin, Gene Nygaard, NekoDaemon, Mets501, CRGreathouse, CBM, Cydebot, Yobot and VladimirReshet-nikov

• Tav (number) Source: https://en.wikipedia.org/wiki/Tav_(number)?oldid=492082102Contributors: Zanimum, Sannse, CharlesMatthews,Maximus Rex, Dbachmann, Ross Burgess, GregorB, Keeves, Koavf, Spacepotato, SmackBot, Imz, JRSpriggs, CBM, Jj137, Chussid,Alksentrs, Hans Adler, Basilicofresco, Yobot, Helpful Pixie Bot and Anonymous: 5

250 CHAPTER 206. Ω-LOGIC

• Teichmüller–Tukey lemma Source: https://en.wikipedia.org/wiki/Teichm%C3%BCller%E2%80%93Tukey_lemma?oldid=688988274Contributors: Michael Hardy, ,דוד Tobias Bergemann, Giftlite, Paul August, Malcolma, Incnis Mrsi, AnOddName, CBM, Marek69,TreasuryTag, Fcady2007, Addbot, Yobot, Erik9bot, HRoestBot, FoxBot, John of Reading, Syberlazer, Brirush, Juanbenitez125 andAnonymous: 2

• The Paradoxes of the Infinite Source: https://en.wikipedia.org/wiki/The_Paradoxes_of_the_Infinite?oldid=699632067 Contributors:Michael Hardy, Trovatore, SmackBot, RDBury, JHunterJ, Trialsanderrors, Gregbard, Thenub314, KConWiki, David Eppstein, Paradoc-tor, Addbot, Zero Thrust, EmausBot, Apollineo!, Mogism, Jochen Burghardt and Mark viking

• Theory of pure equality Source: https://en.wikipedia.org/wiki/Theory_of_pure_equality?oldid=596388457 Contributors: BD2412,CBM and Myasuda

• Trichotomy (mathematics) Source: https://en.wikipedia.org/wiki/Trichotomy_(mathematics)?oldid=682670477Contributors: Zundark,Patrick, Michael Hardy, Arthur Frayn, Casu Marzu, Henrygb, UtherSRG, Tobias Bergemann, Macrakis, Paul August, Oleg Alexandrov,VKokielov, Michael Slone, Bota47, JJL, SmackBot, Mhss, Colonies Chris, Jdthood, Cybercobra, Courcelles, JRSpriggs, Meng.benjamin,Gregbard, Difluoroethene, David Cherney, AntiVandalBot, Indeed123, Minnnnng, YohanN7, Mild Bill Hiccup, MilesAgain, Addbot,Luckas-bot, AnomieBOT, Götz, Xqbot, Constructive editor, ComputScientist, Þjóðólfr, Tkuvho, SporkBot, Paulmiko, Helpful Pixie Bot,Zorglub x and Anonymous: 23

• Truth-table reduction Source: https://en.wikipedia.org/wiki/Truth-table_reduction?oldid=672080295Contributors: MathMartin, EmilJ,Intgr, CharlesHBennett, J. Finkelstein, CBM, Cydebot, DesolateReality, Skeptical scientist, Erik9bot, Mark viking and Anonymous: 4

• Ulammatrix Source: https://en.wikipedia.org/wiki/Ulam_matrix?oldid=674818360 Contributors: R.e.b., Deltahedron, Mark viking andK9re11

• Unfoldable cardinal Source: https://en.wikipedia.org/wiki/Unfoldable_cardinal?oldid=624253226Contributors: Zundark, Schneelocke,Charles Matthews, Dmytro, Tobias Bergemann, Sillydragon, Ben Standeven, Slambo, AndrejBauer, Rjwilmsi, Mets501, JRSpriggs, DavidEppstein, Basilides, Hans Adler, Yobot, Erik9bot, Tkuvho, GoingBatty, Nynj and Anonymous: 3

• Universally Baire set Source: https://en.wikipedia.org/wiki/Universally_Baire_set?oldid=491706639 Contributors: Rjwilmsi, Trovatore,David Eppstein, Hans Adler and Helpful Pixie Bot

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