Inner model theoryFrom Wikipedia, the free encyclopedia

Contents

1 Axiom of constructibility 11.1 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Chang’s model 22.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Code (set theory) 33.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Constructible universe 44.1 What is L? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Additional facts about the sets Lα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 L is a standard inner model of ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.4 L is absolute and minimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.4.1 L and large cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.5 L can be well-ordered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.6 L has a reflection principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.7 The generalized continuum hypothesis holds in L . . . . . . . . . . . . . . . . . . . . . . . . . . 84.8 Constructible sets are definable from the ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.9 Relative constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Continuum hypothesis 115.1 Cardinality of infinite sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Independence from ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Arguments for and against CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 The generalized continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.4.1 Implications of GCH for cardinal exponentiation . . . . . . . . . . . . . . . . . . . . . . . 14

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5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Core model 176.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Construction of core models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Properties of core models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Construction of core models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Covering lemma 197.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 Extenders and indescernibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.4 Additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Easton’s theorem 218.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.2 No extension to singular cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Extender (set theory) 239.1 Formal definition of an extender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Defining an extender from an elementary embedding . . . . . . . . . . . . . . . . . . . . . . . . . 239.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 Goodstein’s theorem 2510.1 Hereditary base-n notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Goodstein sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.3 Proof of Goodstein’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.4 Extended Goodstein’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.5 Sequence length as a function of the starting value . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.6 Application to computable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11 Gödel operation 2911.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

12 Inner model 3112.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.2 Related ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13 Inner model theory 3313.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.2 Consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

14 Jech–Kunen tree 3414.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

15 Jensen hierarchy 3515.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.3 Rudimentary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

16 Kanamori–McAloon theorem 3716.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3716.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

17 Kurepa tree 3817.1 Specializing a Kurepa tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

18 L(R) 4018.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4018.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

19 List of statements undecidable in ZFC 4219.1 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.2 Set theory of the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.3 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.5 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.6 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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19.7 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.8 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.9 Model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4519.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4519.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

20 Minimal model (set theory) 4720.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

21 Mouse (set theory) 4821.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

22 Naimark’s problem 4922.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4922.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

23 Paris–Harrington theorem 5023.1 The strengthened finite Ramsey theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5023.2 The Paris–Harrington theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5023.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5123.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

24 Silver machine 5224.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5224.2 Silver machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5224.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

25 Skolem’s paradox 5425.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5425.2 The paradoxical result and its mathematical implications . . . . . . . . . . . . . . . . . . . . . . . 5425.3 Reception by the mathematical community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5525.4 Current mathematical opinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5625.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5625.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

26 Statements true in L 58

27 Strong measure zero set 5927.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

28 Suslin’s problem 6128.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6128.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6128.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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28.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6228.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

29 Wetzel’s problem 6329.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

30 Whitehead problem 6430.1 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6430.2 Shelah’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6430.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6430.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6530.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

31 Zero sharp 6631.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6631.2 Statements that imply the existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6631.3 Statements equivalent to existence of 0# . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6731.4 Consequences of existence and non-existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6731.5 Other sharps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6731.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6731.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6731.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 69

31.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6931.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7031.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter 1

Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set isconstructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and theconstructible universe, respectively. The axiom, first investigated by Kurt Gödel, is inconsistent with the propositionthat zero sharp exists and stronger large cardinal axioms (see List of large cardinal properties). Generalizations ofthis axiom are explored in inner model theory.

1.1 Implications

The axiom of constructibility implies the axiom of choice over Zermelo–Fraenkel set theory. It also settles manynatural mathematical questions independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Forexample, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin’s hypoth-esis, and the existence of an analytical (in fact,∆1

2 ) non-measurable set of real numbers, all of which are independentof ZFC.The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater orequal to 0#, which includes some “relatively small” large cardinals. Thus, no cardinal can be ω1-Erdős in L. WhileL does contain the initial ordinals of those large cardinals (when they exist in a supermodel of L), and they are stillinitial ordinals in L, it excludes the auxiliary structures (e.g. measures) which endow those cardinals with their largecardinal properties.Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as anaxiom for set theory in the same way as the ZFC axioms. Among set theorists of a realist bent, who believe thatthe axiom of constructibility is either true or false, most believe that it is false. This is in part because it seemsunnecessarily “restrictive”, as it allows only certain subsets of a given set, with no clear reason to believe that theseare all of them. In part it is because the axiom is contradicted by sufficiently strong large cardinal axioms. This pointof view is especially associated with the Cabal, or the “California school” as Saharon Shelah would have it.

1.2 See also• Statements true in L

1.3 References• Devlin, Keith (1984). Constructibility. Springer-Verlag. ISBN 3-540-13258-9.

1.4 External links• How many real numbers are there?, Keith Devlin, Mathematical Association of America, June 2001

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Chapter 2

Chang’s model

In mathematical set theory, Chang’s model is the smallest inner model of set theory closed under countable se-quences. It was introduced by Chang (1971). More generally Chang introduced the smallest inner model closedunder taking sequences of length less than κ for any infinite cardinal κ. For κ countable this is the constructibleuniverse, and for κ the first uncountable cardinal it is Chang’s model.

2.1 References• Chang, C. C. (1971), “Sets constructible using Lκκ", Axiomatic Set Theory, Proc. Sympos. Pure Math., XIII,Part I, Providence, R.I.: Amer. Math. Soc., pp. 1–8, MR 0280357, Zbl 0218.02061

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Chapter 3

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.

3.1 See also• L(R)

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

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Chapter 4

Constructible universe

“Gödel universe” redirects here. For Kurt Gödel’s cosmological solution to the Einstein field equations, see Gödelmetric.

In mathematics, in set theory, the constructible universe (or Gödel’s constructible universe), denoted L, is aparticular class of sets that can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel inhis 1938 paper “The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis”.[1] In this,he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice andthe generalized continuum hypothesis are true in the constructible universe. This shows that both propositions areconsistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold insystems in which one or both of the propositions is true, their consistency is an important result.

4.1 What is L?

L can be thought of as being built in “stages” resembling the von Neumann universe, V. The stages are indexed byordinals. In von Neumann’s universe, at a successor stage, one takes Vα₊₁ to be the set of all subsets of the previousstage, Vα. By contrast, in Gödel’s constructible universe L, one uses only those subsets of the previous stage that are:

• definable by a formula in the formal language of set theory

• with parameters from the previous stage and

• with the quantifiers interpreted to range over the previous stage.

By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resultingsets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory andcontained in any such model.Define

Def(X) :={{y | y ∈ X and (X,∈) |= Φ(y, z1, . . . , zn)}

∣∣∣ Φ and formula first-order a is z1, . . . , zn ∈ X}.

L is defined by transfinite recursion as follows:

• L0 := ∅.

• Lα+1 := Def(Lα).

• If λ is a limit ordinal, then Lλ :=∪

α<λ Lα. Here α<λ means α precedes λ.

• L :=∪

α∈Ord Lα. Here Ord denotes the class of all ordinals.

4

4.2. ADDITIONAL FACTS ABOUT THE SETS LΑ 5

If z is an element of Lα, then z = {y | y ∈ Lα and y ∈ z} ∈ Def (Lα) = Lα₊₁. So Lα is a subset of Lα₊₁, which is asubset of the power set of Lα. Consequently, this is a tower of nested transitive sets. But L itself is a proper class.The elements of L are called “constructible” sets; and L itself is the “constructible universe”. The "axiom of con-structibility", aka “V=L”, says that every set (of V) is constructible, i.e. in L.

4.2 Additional facts about the sets Lα

An equivalent definition for Lα is:

Lα =∪β<α

Def(Lβ)

For any finite ordinal n, the sets L and V are the same (whether V equals L or not), and thus Lω = Vω: theirelements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC inwhich V equals L, Lω₊₁ is a proper subset of Vω₊₁, and thereafter Lα₊₁ is a proper subset of the power set of Lα forall α > ω. On the other hand, V equals L does imply that Vα equals Lα if α = ωα, for example if α is inaccessible.More generally, V equals L implies Hα equals Lα for all infinite cardinals α.If α is an infinite ordinal then there is a bijection between Lα and α, and the bijection is constructible. So these setsare equinumerous in any model of set theory that includes them.As defined above, Def(X) is the set of subsets of X defined by Δ0 formulas (that is, formulas of set theory containingonly bounded quantifiers) that use as parameters only X and its elements.An alternate definition, due to Gödel, characterizes each Lα₊₁ as the intersection of the power set of Lα with theclosure of Lα∪{Lα} under a collection of nine explicit functions. This definition makes no reference to definability.All arithmetical subsets of ω and relations on ω belong to Lω₊₁ (because the arithmetic definition gives one in Lω₊₁).Conversely, any subset of ω belonging to Lω₊₁ is arithmetical (because elements of Lω can be coded by naturalnumbers in such a way that ∈ is definable, i.e., arithmetic). On the other hand, Lω₊₂ already contains certain non-arithmetical subsets of ω, such as the set of (natural numbers coding) true arithmetical statements (this can be definedfrom Lω₊₁ so it is in Lω₊₂).All hyperarithmetical subsets of ω and relations on ω belong to LωCK

1(where ωCK

1 stands for the Church-Kleeneordinal), and conversely any subset of ω that belongs to LωCK

1is hyperarithmetical.[2]

4.3 L is a standard inner model of ZFC

L is a standard model, i.e. it is a transitive class and it uses the real element relationship, so it is well-founded. L isan inner model, i.e. it contains all the ordinal numbers of V and it has no “extra” sets beyond those in V, but it mightbe a proper subclass of V. L is a model of ZFC, which means that it satisfies the following axioms:

• Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.

(L,∈) is a substructure of (V,∈), which is well founded, so L is well founded. In particular, if x∈L, thenby the transitivity of L, y∈L. If we use this same y as in V, then it is still disjoint from x because we areusing the same element relation and no new sets were added.

• Axiom of extensionality: Two sets are the same if and only if they have the same elements.

If x and y are in L and they have the same elements in L, then by L’s transitivity, they have the sameelements (in V). So they are equal (in V and thus in L).

• Axiom of empty set: {} is a set.

6 CHAPTER 4. CONSTRUCTIBLE UNIVERSE

{} = L0 = {y | y∈L0 and y=y} ∈ L1. So {} ∈ L. Since the element relation is the same and no newelements were added, this is the empty set of L.

• Axiom of pairing: If x, y are sets, then {x,y} is a set.

If x∈L and y∈L, then there is some ordinal α such that x∈Lα and y∈Lα. Then {x,y} = {s | s∈Lα and(s=x or s=y)} ∈ Lα₊₁. Thus {x,y} ∈ L and it has the same meaning for L as for V.

• Axiom of union: For any set x there is a set y whose elements are precisely the elements of the elements of x.

If x ∈ Lα, then its elements are in Lα and their elements are also in Lα. So y is a subset of Lα. y = {s |s∈Lα and there exists z∈x such that s∈z} ∈ Lα₊₁. Thus y ∈ L.

• Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.

From transfinite induction, we get that each ordinal α ∈ Lα₊₁. In particular, ω ∈ Lω₊₁ and thus ω ∈ L.

• Axiom of separation: Given any set S and any proposition P(x,z1,...,z ), {x|x∈S and P(x,z1,...,z )} is a set.

By induction on subformulas of P, one can show that there is an α such that Lα contains S and z1,...,zand (P is true in Lα if and only if P is true in L (this is called the "reflection principle")). So {x | x∈S andP(x,z1,...,z ) holds in L} = {x | x∈Lα and x∈S and P(x,z1,...,z ) holds in Lα} ∈ Lα₊₁. Thus the subset isin L.

• Axiom of replacement: Given any set S and anymapping (formally defined as a proposition P(x,y) where P(x,y)and P(x,z) implies y = z), {y | there exists x∈S such that P(x,y)} is a set.

Let Q(x,y) be the formula that relativizes P to L, i.e. all quantifiers in P are restricted to L. Q is a muchmore complex formula than P, but it is still a finite formula, and since P was a mapping over L, Q mustbe a mapping over V; thus we can apply replacement in V to Q. So {y | y∈L and there exists x∈S suchthat P(x,y) holds in L} = {y | there exists x∈S such that Q(x,y)} is a set in V and a subclass of L. Againusing the axiom of replacement in V, we can show that there must be an α such that this set is a subsetof Lα ∈ Lα₊₁. Then one can use the axiom of separation in L to finish showing that it is an element of L.

• Axiom of power set: For any set x there exists a set y, such that the elements of y are precisely the subsets ofx.

In general, some subsets of a set in L will not be in L. So the whole power set of a set in L will usuallynot be in L. What we need here is to show that the intersection of the power set with L is in L. Usereplacement in V to show that there is an α such that the intersection is a subset of Lα. Then theintersection is {z | z∈Lα and z is a subset of x} ∈ Lα₊₁. Thus the required set is in L.

• Axiom of choice: Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containingexactly one element from each member of x.

One can show that there is a definable well-ordering of L which definition works the same way in Litself. So one chooses the least element of each member of x to form y using the axioms of union andseparation in L.

Notice that the proof that L is a model of ZFC only requires that V be a model of ZF, i.e. we do NOT assume thatthe axiom of choice holds in V.

4.4. L IS ABSOLUTE AND MINIMAL 7

4.4 L is absolute and minimal

If W is any standard model of ZF sharing the same ordinals as V, then the L defined in W is the same as the L definedin V. In particular, Lα is the same in W and V, for any ordinal α. And the same formulas and parameters in Def (Lα)produce the same constructible sets in Lα₊₁.Furthermore, since L is a subclass of V and, similarly, L is a subclass of W, L is the smallest class containing all theordinals that is a standard model of ZF. Indeed, L is the intersection of all such classes.If there is a set W in V that is a standard model of ZF, and the ordinal κ is the set of ordinals that occur in W,then Lκ is the L of W. If there is a set that is a standard model of ZF, then the smallest such set is such a Lκ. Thisset is called the minimal model of ZFC. Using the downward Löwenheim–Skolem theorem, one can show that theminimal model (if it exists) is a countable set.Of course, any consistent theory must have a model, so even within the minimal model of set theory there are setsthat are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, theydo not use the normal element relation and they are not well founded.Because both the L of L and the V of L are the real L and both the L of Lκ and the V of Lκ are the real Lκ, we getthat V=L is true in L and in any Lκ that is a model of ZF. However, V=L does not hold in any other standard modelof ZF.

4.4.1 L and large cardinals

Since On⊂L⊆V, properties of ordinals that depend on the absence of a function or other structure (i.e. Π1ZF for-

mulas) are preserved when going down from V to L. Hence initial ordinals of cardinals remain initial in L. Regularordinals remain regular in L. Weak limit cardinals become strong limit cardinals in L because the generalized con-tinuum hypothesis holds in L. Weakly inaccessible cardinals become strongly inaccessible. Weakly Mahlo cardinalsbecome strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinalproperties) will be retained in L.However, 0# is false in L even if true in V. So all the large cardinals whose existence implies 0# cease to have thoselarge cardinal properties, but retain the properties weaker than 0# which they also possess. For example, measurablecardinals cease to be measurable but remain Mahlo in L.Interestingly, if 0# holds in V, then there is a closed unbounded class of ordinals that are indiscernible in L.While someof these are not even initial ordinals in V, they have all the large cardinal properties weaker than 0# in L. Furthermore,any strictly increasing class function from the class of indiscernibles to itself can be extended in a unique way to anelementary embedding of L into L. This gives L a nice structure of repeating segments.

4.5 L can be well-ordered

There are various ways of well-ordering L. Some of these involve the “fine structure” of L, which was first describedby Ronald Bjorn Jensen in his 1972 paper entitled “The fine structure of the constructible hierarchy”. Instead ofexplaining the fine structure, we will give an outline of how L could be well-ordered using only the definition givenabove.Suppose x and y are two different sets in L and we wish to determine whether x<y or x>y. If x first appears in Lα₊₁and y first appears in Lᵦ₊₁ and β is different from α, then let x<y if and only if α<β. Henceforth, we suppose thatβ=α.Remember that Lα₊₁ = Def (Lα), which uses formulas with parameters from Lα to define the sets x and y. If onediscounts (for the moment) the parameters, the formulas can be given a standard Gödel numbering by the naturalnumbers. If Φ is the formula with the smallest Gödel number that can be used to define x, and Ψ is the formula withthe smallest Gödel number that can be used to define y, and Ψ is different from Φ, then let x<y if and only if Φ<Ψin the Gödel numbering. Henceforth, we suppose that Ψ=Φ.Suppose that Φ uses n parameters from Lα. Suppose z1,...,z is the sequence of parameters that can be used with Φto define x, and w1,...,w does the same for y. Then let x<y if and only if either z <w or (z =w and z -₁<w -₁) or(z =w and z -₁=w -₁ and z -₂<w -₂) or etc.. This is called the reverse-lexicographic ordering; if there are multiplesequences of parameters that define one of the sets, we choose the least one under this ordering. It being understood

8 CHAPTER 4. CONSTRUCTIBLE UNIVERSE

that each parameter’s possible values are ordered according to the restriction of the ordering of L to Lα, so thisdefinition involves transfinite recursion on α.The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induc-tion. The values of n-tuples of parameters are well-ordered by the product ordering. The formulas with parametersare well-ordered by the ordered sum (by Gödel numbers) of well-orderings. And L is well-ordered by the orderedsum (indexed by α) of the orderings on Lα₊₁.Notice that this well-ordering can be defined within L itself by a formula of set theory with no parameters, only thefree-variables x and y. And this formula gives the same truth value regardless of whether it is evaluated in L, V, orW (some other standard model of ZF with the same ordinals) and we will suppose that the formula is false if eitherx or y is not in L.It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-orderthe proper class V (as we have done here with L) is equivalent to the axiom of global choice, which is more powerfulthan the ordinary axiom of choice because it also covers proper classes of non-empty sets.

4.6 L has a reflection principle

Proving that the axiom of separation, axiom of replacement, and axiom of choice hold in L requires (at least as shownabove) the use of a reflection principle for L. Here we describe such a principle.By mathematical induction on n<ω, we can use ZF in V to prove that for any ordinal α, there is an ordinal β>α suchthat for any sentence P(z1,...,z ) with z1,...,z in Lᵦ and containing fewer than n symbols (counting a constant symbolfor an element of Lᵦ as one symbol) we get that P(z1,...,z ) holds in Lᵦ if and only if it holds in L.

4.7 The generalized continuum hypothesis holds in L

Let S ∈ Lα , and let T be any constructible subset of S. Then there is some β with T ∈ Lβ+1 , so T = {x ∈Lβ : x ∈ S ∧ Φ(x, zi)} = {x ∈ S : Φ(x, zi)} , for some formula Φ and some zi drawn from Lβ . By thedownward Löwenheim–Skolem theorem, there must be some transitive set K containing Lα and some wi , andhaving the same first-order theory as Lβ with the wi substituted for the zi ; and this K will have the same cardinalas Lα . Since V = L is true in Lβ , it is also true in K, so K = Lγ for some γ having the same cardinal as α. AndT = {x ∈ Lβ : x ∈ S ∧ Φ(x, zi)} = {x ∈ Lγ : x ∈ S ∧ Φ(x,wi)} because Lβ and Lγ have the same theory. SoT is in fact in Lγ+1 .So all the constructible subsets of an infinite set S have ranks with (at most) the same cardinal κ as the rank of S; itfollows that if α is the initial ordinal for κ+, then L∩P(S) ⊆ Lα+1 serves as the “powerset” of S within L. And thisin turn means that the “power set” of S has cardinal at most ||α||. Assuming S itself has cardinal κ, the “power set”must then have cardinal exactly κ+. But this is precisely the generalized continuum hypothesis relativized to L.

4.8 Constructible sets are definable from the ordinals

There is a formula of set theory that expresses the idea that X=Lα. It has only free variables for X and α. Using thiswe can expand the definition of each constructible set. If s∈Lα₊₁, then s = {y|y∈Lα and Φ(y,z1,...,z ) holds in (Lα,∈)}for some formula Φ and some z1,...,z in Lα. This is equivalent to saying that: for all y, y∈s if and only if [there existsX such that X=Lα and y∈X and Ψ(X,y,z1,...,z )] where Ψ(X,...) is the result of restricting each quantifier in Φ(...) toX. Notice that each z ∈Lᵦ₊₁ for some β<α. Combine formulas for the z’s with the formula for s and apply existentialquantifiers over the z’s outside and one gets a formula that defines the constructible set s using only the ordinals α thatappear in expressions like X=Lα as parameters.Example: The set {5,ω} is constructible. It is the unique set, s, that satisfies the formula:∀y(y ∈ s ⇐⇒ (y ∈ Lω+1 ∧ (∀a(a ∈ y ⇐⇒ a ∈ L5 ∧Ord(a)) ∨ ∀b(b ∈ y ⇐⇒ b ∈ Lω ∧Ord(b))))) ,where Ord(a) is short for:∀c ∈ a(∀d ∈ c(d ∈ a ∧ ∀e ∈ d(e ∈ c))).Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would

4.9. RELATIVE CONSTRUCTIBILITY 9

yield. But the point remains, there is a formula of set theory that is true only for the desired constructible set s andthat contains parameters only for ordinals.

4.9 Relative constructibility

Sometimes it is desirable to find a model of set theory that is narrow like L, but that includes or is influenced by aset that is not constructible. This gives rise to the concept of relative constructibility, of which there are two flavors,denoted L(A) and L[A].The class L(A) for a non-constructible set A is the intersection of all classes that are standard models of set theoryand contain A and all the ordinals.L(A) is defined by transfinite recursion as follows:

• L0(A) = the smallest transitive set containing A as an element, i.e. the transitive closure of {A}.

• Lα₊₁(A) = Def (Lα(A))

• If λ is a limit ordinal, then Lλ(A) =∪

α<λ Lα(A) .

• L(A) =∪

α Lα(A) .

If L(A) contains a well-ordering of the transitive closure of {A}, then this can be extended to a well-ordering of L(A).Otherwise, the axiom of choice will fail in L(A).A common example is L(R), the smallest model that contains all the real numbers, which is used extensively inmoderndescriptive set theory.The class L[A] is the class of sets whose construction is influenced by A, where A may be a (presumably non-constructible) set or a proper class. The definition of this class uses DefA (X), which is the same as Def (X) exceptinstead of evaluating the truth of formulas Φ in the model (X,∈), one uses the model (X,∈,A) where A is a unarypredicate. The intended interpretation of A(y) is y∈A. Then the definition of L[A] is exactly that of L only with Defreplaced by DefA.L[A] is always a model of the axiom of choice. Even if A is a set, A is not necessarily itself a member of L[A],although it always is if A is a set of ordinals.It is essential to remember that the sets in L(A) or L[A] are usually not actually constructible and that the propertiesof these models may be quite different from the properties of L itself.

4.10 See also• Axiom of constructibility

• Statements true in L

• Reflection principle

• Axiomatic set theory

• Transitive set

• L(R)

• Ordinal definable

4.11 Notes[1] Gödel, 1938

[2] Barwise 1975, page 60 (comment following proof of theorem 5.9)

10 CHAPTER 4. CONSTRUCTIBLE UNIVERSE

4.12 References• Barwise, Jon (1975). Admissible Sets and Structures. Berlin: Springer-Verlag. ISBN 0-387-07451-1.

• Devlin, Keith J. (1984). Constructibility. Berlin: Springer-Verlag. ISBN 0-387-13258-9.

• Felgner, Ulrich (1971). Models of ZF-Set Theory. Lecture Notes in Mathematics. Springer-Verlag. ISBN3-540-05591-6.

• Gödel, Kurt (1938). “TheConsistency of theAxiom ofChoice and of theGeneralizedContinuum-Hypothesis”.Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sci-ences) 24 (12): 556–557. doi:10.1073/pnas.24.12.556. JSTOR 87239. PMC 1077160. PMID 16577857.

• Gödel, Kurt (1940). The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies 3. Prince-ton, N. J.: Princeton University Press. ISBN 978-0-691-07927-1. MR 0002514.

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

Chapter 5

Continuum hypothesis

This article is about the hypothesis in set theory. For the assumption in fluid mechanics, see Fluid mechanics.

In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the firstof Hilbert’s 23 problems presented in the year 1900. Τhe answer to this problem is independent of ZFC set theory(that is, Zermelo–Fraenkel set theory with the axiom of choice included), so that either the continuum hypothesis orits negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFCis consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in1940.The name of the hypothesis comes from the term the continuum for the real numbers. It is abbreviated CH.

5.1 Cardinality of infinite sets

Main article: Cardinal number

Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspon-dence) between them. Intuitively, for two sets S and T to have the same cardinality means that it is possible to “pairoff” elements of S with elements of T in such a fashion that every element of S is paired off with exactly one elementof T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green} .With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate.The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a propersubset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rationalnumbers than integers, and more real numbers than rational numbers. However, this intuitive analysis does not takeaccount of the fact that all three sets are infinite. It turns out the rational numbers can actually be placed in one-to-onecorrespondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set ofintegers: they are both countable sets.Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers(see Cantor’s first uncountability proof and Cantor’s diagonal argument). His proofs, however, give no indication ofthe extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuumhypothesis as a possible solution to this question.The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinalityof the set of integers. Equivalently, as the cardinality of the integers is ℵ0 ("aleph-naught") and the cardinality of thereal numbers is 2ℵ0 (i.e. it equals the cardinality of the power set of the integers), the continuum hypothesis says thatthere is no set S for which

11

12 CHAPTER 5. CONTINUUM HYPOTHESIS

ℵ0 < |S| < 2ℵ0 .

Assuming the axiom of choice, there is a smallest cardinal number ℵ1 greater than ℵ0 , and the continuum hypothesisis in turn equivalent to the equality

2ℵ0 = ℵ1.

A consequence of the continuum hypothesis is that every infinite subset of the real numbers either has the samecardinality as the integers or the same cardinality as the entire set of the reals.There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH)which says that for all ordinals α

2ℵα = ℵα+1.

That is, GCH asserts that the cardinality of the power set of any infinite set is the smallest cardinality greater thanthat of the set.

5.2 Independence from ZFC

Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain (Dauben 1990). Itbecame the first on David Hilbert’s list of important open questions that was presented at the International Congressof Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated.Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standardZermelo–Fraenkel set theory (ZF), even if the axiom of choice is adopted (ZFC) (Gödel (1940)). Paul Cohen showedin 1963 that CH cannot be proven from those same axioms either (Cohen (1963) & Cohen (1964)). Hence, CH isindependent of ZFC. Both of these results assume that the Zermelo–Fraenkel axioms are consistent; this assumptionis widely believed to be true. Cohen was awarded the Fields Medal in 1966 for his proof.The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory.As a result of its independence, many substantial conjectures in those fields have subsequently been shown to beindependent as well.So far, CH appears to be independent of all known large cardinal axioms in the context of ZFC. (Feferman (1999))The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödeland Cohen’s negative results are not universally accepted as disposing of the hypothesis. Hilbert’s problem remainsan active topic of research; see Woodin (2001) and Koellner (2011a) for an overview of the current research status.The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequenceof Gödel’s incompleteness theorem, which was published in 1931, is that there is a formal statement (one for eachappropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC, assuming thatZFC is consistent. The continuum hypothesis and the axiom of choice were among the first mathematical statementsshown to be independent of ZF set theory. These proofs of independence were not completed until Paul Cohendeveloped forcing in the 1960s. They all rely on the assumption that ZF is consistent. These proofs are called proofsof relative consistency (see Forcing (mathematics)).A result of Solovay, proved shortly after Cohen’s result on the independence of the continuum hypothesis, shows thatin any model of ZFC, if κ is a cardinal of uncountable cofinality, then there is a forcing extension in which 2ℵ0 = κ. However, it is not consistent to assume 2ℵ0 is ℵω or ℵω1+ω or any cardinal with cofinality ω .

5.3 Arguments for and against CH

Gödel believed that CH is false and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a platonist and therefore had no

5.4. THE GENERALIZED CONTINUUM HYPOTHESIS 13

problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though aformalist (Goodman 1979), also tended towards rejecting CH.Historically, mathematicians who favored a “rich” and “large” universe of sets were against CH, while those favoringa “neat” and “controllable” universe favored CH. Parallel arguments were made for and against the axiom of con-structibility, which implies CH. More recently, Matthew Foreman has pointed out that ontological maximalism canactually be used to argue in favor of CH, because among models that have the same reals, models with “more” setsof reals have a better chance of satisfying CH (Maddy 1988, p. 500).Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. Thisviewpoint was advanced as early as 1923 by Skolem, even beforeGödel’s first incompleteness theorem. Skolem arguedon the basis of what is now known as Skolem’s paradox, and it was later supported by the independence of CH fromthe axioms of ZFC, since these axioms are enough to establish the elementary properties of sets and cardinalities. Inorder to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuitionand resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generallyconsidered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171).At least two other axioms have been proposed that have implications for the continuum hypothesis, although theseaxioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presentedan argument against CH by showing that the negation of CH is equivalent to Freiling’s axiom of symmetry, a statementabout probabilities. Freiling believes this axiom is “intuitively true” but others have disagreed. A difficult argumentagainst CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000 (Woodin 2001a,2001b). Foreman (2003) does not reject Woodin’s argument outright but urges caution.Solomon Feferman (2011) has made a complex philosophical argument that CH is not a definite mathematical prob-lem. He proposes a theory of “definiteness” using a semi-intuitionistic subsystem of ZF that accepts classical logic forbounded quantifiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition ϕ is mathemati-cally “definite” if the semi-intuitionistic theory can prove (ϕ∨¬ϕ) . He conjectures that CH is not definite accordingto this notion, and proposes that CH should therefore be considered not to have a truth value. Peter Koellner (2011b)wrote a critical commentary on Feferman’s article.Joel David Hamkins proposes a multiverse approach to set theory and argues that “the continuum hypothesis is settledon the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can nolonger be settled in the manner formerly hoped for.” (Hamkins 2012). In a related vein, Saharon Shelah wrote thathe does “not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we justhave to discover the additional axiom. My mental picture is that we have many possible set theories, all conformingto ZFC.” (Shelah 2003).

5.4 The generalized continuum hypothesis

The generalized continuum hypothesis (GCH) states that if an infinite set’s cardinality lies between that of an infiniteset S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as thepower set of S. That is, for any infinite cardinal λ there is no cardinal κ such that λ < κ < 2λ. GCH is equivalentto:

ℵα+1 = 2ℵα for every ordinal α. (occasionally called Cantor’s aleph hypothesis)

The beth numbers provide an alternate notation for this condition: ℵα = ℶα for every ordinal α.This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power setof the integers. It was first suggested by Jourdain (1905).Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC),so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. Toprove this, Sierpiński showed GCH implies that every cardinality n is smaller than some Aleph number, and thus canbe ordered. This is done by showing that n is smaller than 2ℵ0+n which is smaller than its own Hartogs number —this uses the equality 2ℵ0+n = 2 · 2ℵ0+n ; for the full proof, see Gillman (2002).Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to theordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen’s model in which CH fails is a modelin which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed

14 CHAPTER 5. CONTINUUM HYPOTHESIS

by Cohen to prove Easton’s theorem, which shows it is consistent with ZFC for arbitrarily large cardinals ℵα tofail to satisfy 2ℵα = ℵα+1. Much later, Foreman and Woodin proved that (assuming the consistency of very largecardinals) it is consistent that 2κ > κ+ holds for every infinite cardinal κ. Later Woodin extended this by showingthe consistency of 2κ = κ++ for every κ . A recent result of Carmi Merimovich shows that, for each n≥1, it isconsistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, László Patai (1930) proved, thatif γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite.For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsetsof B. Thus for any infinite cardinals A and B,

A < B → 2A ≤ 2B.

If A and B are finite, the stronger inequality

A < B → 2A < 2B

holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.

5.4.1 Implications of GCH for cardinal exponentiation

Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, onecan deduce from it the values of cardinal exponentiation in all cases. It implies that ℵℵβ

α is (see: Hayden & Kennison(1968), page 147, exercise 76):

ℵβ+1 when α ≤ β+1;ℵα when β+1 < α and ℵβ < cf(ℵα) where cf is the cofinality operation; andℵα+1 when β+1 < α and ℵβ ≥ cf(ℵα) .

5.5 See also

• Aleph number

• Beth number

• Cardinality

• Ω-logic

• Wetzel’s problem

5.6 References

• Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: DoverPublications. ISBN 978-0-486-46921-8.

• Cohen, Paul J. (December 15, 1963). “The Independence of the Continuum Hypothesis”. Proceedings of theNational Academy of Sciences of the United States of America 50 (6): 1143–1148. doi:10.1073/pnas.50.6.1143.JSTOR 71858. PMC 221287. PMID 16578557.

• Cohen, Paul J. (January 15, 1964). “The Independence of the Continuum Hypothesis, II”. Proceedings of theNational Academy of Sciences of the United States of America 51 (1): 105–110. doi:10.1073/pnas.51.1.105.JSTOR 72252. PMC 300611. PMID 16591132.

• Dales, H. G.; Woodin, W. H. (1987). An Introduction to Independence for Analysts. Cambridge.

5.6. REFERENCES 15

• Dauben, Joseph Warren (1990). Georg Cantor: His Mathematics and Philosophy of the Infinite. PrincetonUniversity Press. pp. 134–137. ISBN 9780691024479.

• Enderton, Herbert (1977). Elements of Set Theory. Academic Press.

• Feferman, Solomon (February 1999). “Doesmathematics need new axioms?". AmericanMathematicalMonthly106 (2): 99–111. doi:10.2307/2589047.

• Feferman, Solomon (2011). “Is the ContinuumHypothesis a definitemathematical problem?" (PDF).Exploringthe Frontiers of Independence (Harvard lecture series). External link in |work= (help)

• Foreman, Matt (2003). “Has the Continuum Hypothesis been Settled?" (PDF). Retrieved February 25, 2006.

• Freiling, Chris (1986). “Axioms of Symmetry: Throwing Darts at the Real Number Line”. Journal of SymbolicLogic (Association for Symbolic Logic) 51 (1): 190–200. doi:10.2307/2273955. JSTOR 2273955.

• Gödel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press.

• Gillman, Leonard (2002). “Two Classical Surprises Concerning the Axiom of Choice and the ContinuumHypothesis” (PDF). American Mathematical Monthly 109. doi:10.2307/2695444.

• Gödel, K.: What is Cantor’s Continuum Problem?, reprinted in Benacerraf and Putnam’s collection Philosophyof Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel’s arguments against CH.

• Goodman, Nicolas D. (1979). “Mathematics as an objective science”. The American Mathematical Monthly86 (7): 540–551. doi:10.2307/2320581. MR 542765. This view is often called formalism. Positions more orless like this may be found in Haskell Curry [5], Abraham Robinson [17], and Paul Cohen [4].

• Joel David Hamkins. The set-theoretic multiverse. Rev. Symb. Log. 5 (2012), no. 3, 416–449.

• Seymour Hayden and John F. Kennison: Zermelo–Fraenkel Set Theory (1968), Charles E. Merrill PublishingCompany, Columbus, Ohio.

• Jourdain, Philip E. B. (1905). “On transfinite cardinal numbers of the exponential form”. Philosophical Mag-azine, Series 6 9: 42–56. doi:10.1080/14786440509463254.

• Koellner, Peter (2011a). “The Continuum Hypothesis” (PDF). Exploring the Frontiers of Independence (Har-vard lecture series).

• Koellner, Peter (2011b). “Feferman On the Indefiniteness of CH” (PDF).

• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Amsterdam: North-Holland.ISBN 978-0-444-85401-8.

• Maddy, Penelope (June 1988). “Believing the Axioms, I”. Journal of Symbolic Logic (Association for SymbolicLogic) 53 (2): 481–511. doi:10.2307/2274520. JSTOR 2274520.

• Martin, D. (1976). “Hilbert’s first problem: the continuum hypothesis,” inMathematical Developments ArisingfromHilbert’s Problems, Proceedings of Symposia in PureMathematics XXVIII, F. Browder, editor. AmericanMathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1

• McGough, Nancy. “The Continuum Hypothesis”.

• Merimovich, Carmi (2007). “A power function with a fixed finite gap everywhere”. Journal of Symbolic Logic72 (2): 361–417. doi:10.2178/jsl/1185803615.

• Moore, Gregory H. (2011). “Early history of the generalized continuum hypothesis: 1878–1938”. Bull. Sym-bolic Logic 17 (4): 489–532. doi:10.2178/bsl/1318855631. MR 2896574.

• Shelah, Saharon (2003). “Logical dreams”. Bull. Amer. Math. Soc. (N.S.) 40 (2): 203–228. doi:10.1090/s0273-0979-03-00981-9.

• Woodin, W. Hugh (2001a). “The Continuum Hypothesis, Part I” (PDF). Notices of the AMS 48 (6): 567–576.

• Woodin, W. Hugh (2001b). “The ContinuumHypothesis, Part II” (PDF).Notices of the AMS 48 (7): 681–690.

16 CHAPTER 5. CONTINUUM HYPOTHESIS

German literature

• Cantor, Georg (1878). “Ein Beitrag zur Mannigfaltigkeitslehre”. Journal für die Reine und Angewandte Math-ematik 84: 242–258. doi:10.1515/crll.1878.84.242.

• Patai, L. (1930). “Untersuchungen über die .”reihe-א Mathematische und naturwissenschaftliche Berichte ausUngarn 37: 127–142.

5.7 External links• Szudzik, Matthew and Weisstein, Eric W., “Continuum Hypothesis”, MathWorld.

This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

Chapter 6

Core model

In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to“the core model”, it is not a uniquely identified mathematical object. Rather, it is a class of inner models that underthe right set theoretic assumptions have very special properties, most notably covering properties. Intuitively, thecore model is “the largest canonical inner model there is” (Ernest Schimmerling and John R. Steel) and is typicallyassociated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase “core model below Φ" refers tothe definable inner model that exhibits the special properties under the assumption that there does not exist a cardinalsatisfying Φ. The core model program seeks to analyze large cardinal axioms by determining the core models belowthem.

6.1 History

The first core model was Kurt Gödel's constructible universe L. Ronald Jensen proved the covering lemma for L inthe 1970s under the assumption of the non-existence of zero sharp, establishing that L is the “core model below zerosharp”. The work of Solovay isolated another core model L[U], for U an ultrafilter on a measurable cardinal (and itsassociated “sharp”, zero dagger). Together with Tony Dodd, Jensen constructed the Dodd–Jensen core model (“thecore model below a measurable cardinal”) and proved the covering lemma for it and a generalized covering lemmafor L[U].Mitchell used coherent sequences of measures to develop core models containing multiple or higher-order measur-ables. Still later, the Steel core model used extenders and iteration trees to construct a core model below a Woodincardinal.

6.2 Construction of core models

Core models are constructed by transfinite recursion from small fragments of the core model called mice. An im-portant ingredient of the construction is the comparison lemma that allows giving a wellordering of the relevantmice.At the level of strong cardinals and above, one constructs an intermediate countably certified core model Kc, andthen, if possible, extracts K from Kc.

6.3 Properties of core models

K (and hence K) is a fine-structural countably iterable extender model below long extenders. (It is not currentlyknown how to deal with long extenders, which establish that a cardinal is superstrong.) Here countable iterabilitymeans ω1+1 iterability for all countable elementary substructures of initial segments, and it suffices to develop basictheory, including certain condensation properties. The theory of such models is canonical and well-understood. Theysatisfy GCH, the diamond principle for all stationary subsets of regular cardinals, the square principle (except atsubcompact cardinals), and other principles holding in L.

17

18 CHAPTER 6. CORE MODEL

Kc is maximal in several senses. Kc computes the successors of measurable and many singular cardinals correctly.Also, it is expected that under an appropriate weakening of countable certifiability, Kc would correctly compute thesuccessors of all weakly compact and singular strong limit cardinals correctly. If V is closed under a mouse operator(an inner model operator), then so is Kc. Kc has no sharp: There is no natural non-trivial elementary embedding ofKc into itself. (However, unlike K, Kc may be elementarily self-embeddable.)If in addition there are also noWoodin cardinals in this model (except in certain specific cases, it is not known how thecore model should be defined if K has Woodin cardinals), we can extract the actual core model K. K is also its owncore model. K is locally definable and generically absolute: For every generic extension of V, for every cardinal κ>ω1

in V[G], K as constructed in H(κ) of V[G] equals K∩H(κ). (This would not be possible had K contained Woodincardinals). K is maximal, universal, and fully iterable. This implies that for every iterable extender model M (calleda mouse), there is an elementary embedding M→N and of an initial segment of K into N, and if M is universal, theembedding is of K into M.It is conjectured that if K exists and V is closed under a sharp operator M, then K is Σ1

1 correct allowing real numbersin K as parameters and M as a predicate. That amounts to Σ1

3 correctness (in the usual sense) if M is x→x#.The core model can also be defined above a particular set of ordinals X: X belongs to K(X), but K(X) satisfies theusual properties of K above X. If there is no iterable inner model with ω Woodin cardinals, then for some X, K(X)exists. The above discussion of K and Kc generalizes to K(X) and Kc(X).

6.4 Construction of core models

Conjecture:

• If there is no ω1+1 iterable model with long extenders (and hence models with superstrong cardinals), then Kc

exists.

• If Kc exists and as constructed in every generic extension of V (equivalently, under some generic collapseColl(ω, <κ) for a sufficiently large ordinal κ) satisfies “there are no Woodin cardinals”, then the Core Model Kexists.

Partial results for the conjecture are that:

1. If there is no inner model with a Woodin cardinal, then K exists.

2. If (boldface) Σ1 determinacy (n is finite) holds in every generic extension of V, but there is no iterable innermodel with n Woodin cardinals, then K exists.

3. If there is a measurable cardinal κ, then either Kc below κ exists, or there is an ω1+1 iterable model withmeasurable limit λ of both Woodin cardinals and cardinals strong up to λ.

If V has Woodin cardinals but not cardinals strong past a Woodin one, then under appropriate circumstances (acandidate for) K can be constructed by constructing K below eachWoodin cardinal (and below the class of all ordinals)κ but above that K as constructed below the supremum of Woodin cardinals below κ. The candidate core model isnot fully iterable (iterability fails at Woodin cardinals) or generically absolute, but otherwise behaves like K.

6.5 References• W.H. Woodin (2001). The Continuum Hypothesis, Part I. Notices of the AMS.

• William Mitchell. “Beginning Inner Model Theory” (being Chapter 17 in Volume 3 of “Handbook of SetTheory”) at .

• Matthew Foreman and Akihiro Kanamori (Editors). “Handbook of Set Theory”, Springer Verlag, 2010, ISBN978-1402048432.

• Ronald Jensen and John Steel. “K without the measurable”. Journal of Symbolic Logic Volume 78, Issue 3(2013), 708-734.

Chapter 7

Covering lemma

See also: Jensen’s covering theorem

In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinalsleads to the existence of a canonical inner model, called the core model, that is, in a sense, maximal and approximatesthe structure of the vonNeumann universeV. A covering lemma asserts that under some particular anti-large cardinalassumption, the core model exists and is maximal in a sense that depends on the chosen large cardinal.

7.1 Example

For example, if there is no inner model for a measurable cardinal, then the Dodd–Jensen core model, KDJ is the coremodel and satisfies the covering property, that is for every uncountable set x of ordinals, there is y such that y⊃x, yhas the same cardinality as x, and y ∈KDJ. (If 0# does not exist, then KDJ=L.)

7.2 Versions

If the core model K exists (and has no Woodin cardinals), then

1. If K has no ω1-Erdős cardinals, then for a particular countable (in K) and definable in K sequence of functionsfrom ordinals to ordinals, every set of ordinals closed under these functions is a union of a countable numberof sets in K. If L=K, these are simply the primitive recursive functions.

2. If K has no measurable cardinals, then for every uncountable set x of ordinals, there is y∈K such that x ⊂ y and|x|=|y|.

3. If K has only one measurable cardinal κ, then for every uncountable set x of ordinals, there is y∈K[C] suchthat x ⊂ y and |x|=|y|. Here C is either empty or Prikry generic over K (so it has order type ω and is cofinal inκ) and unique except up to a finite initial segment.

4. If K has no inaccessible limit of measurable cardinals and no proper class of measurable cardinals, then thereis a maximal and unique (except for a finite set of ordinals) set C (called a system of indiscernibles) for K suchthat for every sequence S in K of measure one sets consisting of one set for each measurable cardinal, C minus∪S is finite. Note that every κ\C is either finite or Prikry generic for K at κ except for members of C belowa measurable cardinal below κ. For every uncountable set x of ordinals, there is y∈K[C] such that x ⊂ y and|x|=|y|.

5. For every uncountable set x of ordinals, there is a set C of indiscernibles for total extenders on K such thatthere is y∈K[C] and x ⊂ y and |x|=|y|.

6. K computes the successors of singular and weakly compact cardinals correctly (Weak Covering Property).Moreover, if |κ|>ω1, then cofinality((κ+)K ) ≥ |κ|.

19

20 CHAPTER 7. COVERING LEMMA

7.3 Extenders and indescernibles

For core models without overlapping total extenders, the systems of indescernibles are well-understood. Although(if K has an inaccessible limit of measurable cardinals), the system may depend on the set to be covered, it is well-determined and unique in a weaker sense. One application of the covering is counting the number of (sequencesof) indiscernibles, which gives optimal lower bounds for various failures of the singular cardinals hypothesis. Forexample, if K does not have overlapping total extenders, and κ is singular strong limit, and 2κ=κ++, then κ hasMitchell order at least κ++ in K. Conversely, a failure of the singular cardinal hypothesis can be obtained (in a genericextension) from κ with o(κ)=κ++.For core models with overlapping total extenders (that is with a cardinal strong up to a measurable one), the systemsof indiscernibles are poorly understood, and applications (such as the weak covering) tend to avoid rather than analyzethe indiscernibles.

7.4 Additional properties

If K exists, then every regular Jónsson cardinal is Ramsey in K. Every singular cardinal that is regular in K is mea-surable in K.Also, if the core model K(X) exists above a set X of ordinals, then it has the above discussed covering propertiesabove X.

7.5 References• 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

Chapter 8

Easton’s theorem

In set theory, Easton’s theorem is a result on the possible cardinal numbers of powersets. Easton (1970) (extendinga result of Robert M. Solovay) showed via forcing that

κ < cf(2κ)

and, for κ < λ , that

2κ ≤ 2λ

are the only constraints on permissible values for 2κ when κ is a regular cardinal.

8.1 Statement of the theorem

Easton’s theorem states that if G is a class function whose domain consists of ordinals and whose range consists ofordinals such that

1. G is non-decreasing,

2. the cofinality of ℵG(α) is greater than ℵα for each α in the domain of G, and

3. ℵα is regular for each α in the domain of G,

then there is a model of ZFC such that

2ℵα = ℵG(α)

for each α in the domain of G.The proof of Easton’s theorem uses forcing with a proper class of forcing conditions over a model satisfying thegeneralized continuum hypothesis.The first two conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, whilecondition 2 follows from König’s theorem.In Easton’s model the powersets of singular cardinals have the smallest possible cardinality compatible with theconditions that 2κ has cofinality greater than κ and is a non-decreasing function of κ.

21

22 CHAPTER 8. EASTON’S THEOREM

8.2 No extension to singular cardinals

Silver (1975) proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which thegeneralized continuum hypothesis fails. This shows that Easton’s theorem cannot be extended to the class of allcardinals. The program of PCF theory gives results on the possible values of 2λ for singular cardinals λ . PCF theoryshows that the values of the continuum function on singular cardinals are strongly influenced by the values on smallercardinals, whereas Easton’s theorem shows that the values of the continuum function on regular cardinals are onlyweakly influenced by the values on smaller cardinals.

8.3 See also• Singular cardinal hypothesis

• Aleph number

• Beth number

8.4 References• Easton, W. (1970), “Powers of regular cardinals”, Ann. Math. Logic 1 (2): 139–178, doi:10.1016/0003-4843(70)90012-4

• Silver, Jack (1975), “On the singular cardinals problem”, Proceedings of the International Congress of Mathe-maticians (Vancouver, B. C., 1974) 1, Montreal, Que.: Canad. Math. Congress, pp. 265–268, MR 0429564

Chapter 9

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.

9.1 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

9.2 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:

23

24 CHAPTER 9. EXTENDER (SET THEORY)

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.

9.3 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 10

Goodstein’s theorem

“hydra game” redirects here. For the game development kit, see HYDRA Game Development Kit.

In mathematical logic,Goodstein’s theorem is a statement about the natural numbers, proved by Reuben Goodsteinin 1944, which states that every Goodstein sequence eventually terminates at 0. Kirby and Paris[1] showed that it isunprovable in Peano arithmetic (but it can be proven in stronger systems, such as second order arithmetic). This wasthe third example of a true statement that is unprovable in Peano arithmetic, after Gödel’s incompleteness theorem andGerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harringtontheorem was a later example.Laurence Kirby and Jeff Paris introduced a graph theoretic hydra game with behavior similar to that of Goodsteinsequences: the "Hydra" is a rooted tree, and a move consists of cutting off one of its “heads” (a branch of the tree), towhich the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris provedthat the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though thismay take a very long time.[1]

10.1 Hereditary base-n notation

Goodstein sequences are defined in terms of a concept called “hereditary base-n notation”. This notation is verysimilar to usual base-n positional notation, but the usual notation does not suffice for the purposes of Goodstein’stheorem.In ordinary base-n notation, where n is a natural number greater than 1, an arbitrary natural number m is written asa sum of multiples of powers of n:

m = aknk + ak−1n

k−1 + · · ·+ a0,

where each coefficient ai satisfies 0 ≤ ai < n, and ak ≠ 0. For example, in base 2,

35 = 32 + 2 + 1 = 25 + 21 + 20.

Thus the base 2 representation of 35 is 100011, which means 25 + 2 + 1. Similarly, 100 represented in base 3 is10201:

100 = 81 + 18 + 1 = 34 + 2 · 32 + 30.

Note that the exponents themselves are not written in base-n notation. For example, the expressions above include25 and 34.To convert a base-n representation to hereditary base n notation, first rewrite all of the exponents in base-n notation.Then rewrite any exponents inside the exponents, and continue in this way until every digit appearing in the expressionis n or less.

25

26 CHAPTER 10. GOODSTEIN’S THEOREM

For example, while 35 in ordinary base-2 notation is 25 + 2 + 1, it is written in hereditary base-2 notation as

35 = 222+1 + 2 + 1,

using the fact that 5 = 22 + 1. Similarly, 100 in hereditary base 3 notation is

100 = 33+1 + 2 · 32 + 1.

10.2 Goodstein sequences

The Goodstein sequence G(m) of a number m is a sequence of natural numbers. The first element in the sequenceG(m) is m itself. To get the second, G(m)(2), write m in hereditary base 2 notation, change all the 2s to 3s, and thensubtract 1 from the result. In general, the n+1st term G(m)(n+1) of the Goodstein sequence of m is as follows: takethe hereditary base n+1 representation of G(m)(n), and replace each occurrence of the base n+1 with n+2 and thensubtract one. Note that the next term depends both on the previous term and on the index n. Continue until the resultis zero, at which point the sequence terminates.Early Goodstein sequences terminate quickly. For example, G(3) terminates at the sixth step:

Later Goodstein sequences increase for a very large number of steps. For example,G(4) A056193 starts as follows:Elements of G(4) continue to increase for a while, but at base 3 · 2402653209 , they reach the maximum of 3 ·2402653210 − 1 , stay there for the next 3 · 2402653209 steps, and then begin their first and final descent.The value 0 is reached at base 3 · 2402653211 − 1 . (Curiously, this is a Woodall number: 3 · 2402653211 − 1 =402653184 · 2402653184 − 1 . This is also the case with all other final bases for starting values greater than 4.)However, even G(4) doesn't give a good idea of just how quickly the elements of a Goodstein sequence can increase.G(19) increases much more rapidly, and starts as follows:In spite of this rapid growth, Goodstein’s theorem states that every Goodstein sequence eventually terminates at0, no matter what the starting value is.

10.3 Proof of Goodstein’s theorem

Goodstein’s theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Good-stein sequence G(m), we construct a parallel sequence P(m) of ordinal numbers which is strictly decreasing and ter-minates. Then G(m) must terminate too, and it can terminate only when it goes to 0. A common misunderstandingof this proof is to believe that G(m) goes to 0 because it is dominated by P(m). In fact that P(m) dominates G(m)plays no role at all. The important points is: G(m)(k) exists if and only if P(m)(k) exists (parallelism). Then if P(m)terminates, so does G(m). And G(m) can terminate only when it comes to 0.More precisely, each term P(m)(n) of the sequence P(m) is obtained by applying a function f on the term G(m)(n)of the Goodstein sequence of m as follows: take the hereditary base n+1 representation of G(m)(n), and replace eachoccurrence of the base n+1 with the first infinite ordinal number ω. For example G(3)(1) = 3 = 21 + 20 and P(3)(1)= f(G(3)(1)) = ω1 + ω0 = ω + 1. Addition, multiplication and exponentiation of ordinal numbers are well defined.

• The base-changing operation of the Goodstein sequence when going from G(m)(n) to G(m)(n+1) does notchange the value of f (that’s the main point of the construction), thus after the minus 1 operation, P(m)(n+1)will be strictly smaller than P(m)(n). For example, f(3 · 444 + 4) = 3ωωω

+ ω = f(3 · 555 + 5) , hencef(3 · 444 + 4) is strictly greater than f((3 · 555 + 5)− 1).

If the sequence G(m) did not go to 0, it would not terminate and would be infinite (since G(m)(k+1) would alwaysexist). Consequently, P(m) also would be infinite (since in its turn P(m)(k+1) would always exist too). But P(m)is strictly decreasing and the standard order < on ordinals is well-founded, therefore an infinite strictly decreasingsequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals does terminate (and cannot be

10.4. EXTENDED GOODSTEIN’S THEOREM 27

infinite). This contradiction shows that G(m) terminates, and since it terminates, goes to 0 (by the way, since thereexists a natural number k such that G(m)(k) = 0, by construction of P(m) we have that P(m)(k) = 0).While this proof of Goodstein’s theorem is fairly easy, the Kirby–Paris theorem,[1] which shows that Goodstein’stheorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countablenonstandard models of Peano arithmetic. What Kirby showed is that Goodstein’s theorem leads to Gentzen’s theorem,i.e. it can substitute for induction up to ε0.

10.4 Extended Goodstein’s theorem

Suppose the definition Goodstein sequence is changed so that instead of replacing each occurrence of the base bwith b+1 it was replaces it with b+2. Would the sequence still terminate? More generally, let b1, b2, b3, … beany sequences of integers. Then let the n+1st term G(m)(n+1) of the extended Goodstein sequence of m be asfollows: take the hereditary base bn representation of G(m)(n), and replace each occurrence of the base bn withbn+1 and then subtract one. The claim is that this sequence still terminates. The extended proof defines P(m)(n)= f(G(m)(n), n) as follows: take the hereditary base bn representation of G(m)(n), and replace each occurrence ofthe base bn with the first infinite ordinal number ω. The base-changing operation of the Goodstein sequence whengoing from G(m)(n) to G(m)(n+1) still does not change the value of f. For example, if bn = 4 and if bn+1 = 9, thenf(3 · 444 + 4, 4) = 3ωωω

+ ω = f(3 · 999 + 9, 9) , hence the ordinal f(3 · 444 + 4, 4) is strictly greater than theordinal f((3 · 999 + 9)− 1, 9).

10.5 Sequence length as a function of the starting value

The Goodstein function, G : N → N , is defined such that G(n) is the length of the Goodstein sequence that startswith n. (This is a total function since every Goodstein sequence terminates.) The extreme growth-rate of G can becalibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functionsHα in theHardy hierarchy, and the functions fα in the fast-growing hierarchy of Löb and Wainer:

• Kirby and Paris (1982) proved that

G has approximately the same growth-rate asHϵ0 (which is the same as that of fϵ0 ); more precisely, Gdominates Hα for every α < ϵ0 , andHϵ0 dominates G.(For any two functions f, g : N → N , f is said to dominate g if f(n) > g(n) for all sufficiently largen .)

• Cichon (1983) showed that

G(n) = HRω2 (n+1)(1)− 1,

where Rω2 (n) is the result of putting n in hereditary base-2 notation and then replacing all 2s with ω (as

was done in the proof of Goodstein’s theorem).

• Caicedo (2007) showed that if n = 2m1 + 2m2 + · · ·+ 2mk withm1 > m2 > · · · > mk, then

G(n) = fRω2 (m1)(fRω

2 (m2)(· · · (fRω2 (mk)(3)) · · · ))− 2

Some examples:(For Ackermann function and Graham’s number bounds see fast-growing hierarchy#Functions in fast-growing hier-archies.)

10.6 Application to computable functions

Goodstein’s theorem can be used to construct a total computable function that Peano arithmetic cannot prove to betotal. The Goodstein sequence of a number can be effectively enumerated by a Turing machine; thus the function

28 CHAPTER 10. GOODSTEIN’S THEOREM

which maps n to the number of steps required for the Goodstein sequence of n to terminate is computable by aparticular Turing machine. This machine merely enumerates the Goodstein sequence of n and, when the sequencereaches 0, returns the length of the sequence. Because every Goodstein sequence eventually terminates, this functionis total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmeticdoes not prove that this Turing machine computes a total function.

10.7 See also• Non-standard model of arithmetic

• Fast-growing hierarchy

• Paris–Harrington theorem

• Kanamori–McAloon theorem

• Kruskal’s tree theorem

10.8 References[1] Kirby, L.; Paris, J. (1982). “Accessible Independence Results for Peano Arithmetic” (PDF). Bulletin of the London Math-

ematical Society 14 (4): 285. doi:10.1112/blms/14.4.285.

10.9 Bibliography• Goodstein, R. (1944), “On the restricted ordinal theorem”, Journal of Symbolic Logic 9: 33–41, doi:10.2307/2268019,JSTOR 2268019.

• Cichon, E. (1983), “A Short Proof of Two Recently Discovered Independence Results Using Recursive The-oretic Methods”, Proceedings of the American Mathematical Society 87: 704–706, doi:10.2307/2043364,JSTOR 2043364.

• Caicedo, A. (2007), “Goodstein’s function” (PDF), Revista Colombiana de Matemáticas 41 (2): 381–391.

10.10 External links• Weisstein, Eric W., “Goodstein Sequence”, MathWorld.

• Some elements of a proof that Goodstein’s theorem is not a theorem of PA, from an undergraduate thesis byJustin T Miller

• A Classification of non standard models of Peano Arithmetic by Goodstein’s theorem - Thesis by Dan Kaplan,Franklan and Marshall College Library

• Definitions of Goodstein sequences in the programming languages Ruby and Haskell, as well as a large-scaleplot

• The Hydra game implemented as a Java applet

• Goodstein Sequences: The Power of a Detour via Infinity - good exposition with illustrations of GoodsteinSequences and the hydra game.

Chapter 11

Gödel operation

In mathematical set theory, a set of Gödel operations is a finite collection of operations on sets that can be usedto construct the constructible sets from ordinals. Gödel (1940) introduced the original set of 8 Gödel operations𝔉1,...,𝔉8 under the name fundamental operations. Other authors sometimes use a slightly different set of about 8to 10 operations, usually denoted G1, G2,...

11.1 Definition

Gödel (1940) used the following eight operations as a set of Gödel operations (which he called fundamental opera-tions):

1. F1(X,Y ) = {X,Y }

2. F2(X,Y ) = E ·X = {(a, b) ∈ X | a ∈ b}

3. F3(X,Y ) = X − Y

4. F4(X,Y ) = X ↾ Y = X · (V × Y ) = {(a, b) ∈ X | b ∈ Y }

5. F5(X,Y ) = X ·D(Y ) = {b ∈ X | ∃a(a, b) ∈ Y }

6. F6(X,Y ) = X · Y −1 = {(a, b) ∈ X | (b, a) ∈ Y }

7. F7(X,Y ) = X · Cnv2(Y ) = {(a, b, c) ∈ X | (a, c, b) ∈ Y }

8. F8(X,Y ) = X · Cnv3(Y ) = {(a, b, c) ∈ X | (c, a, b) ∈ Y }

The second expression in each line gives Gödel’s definition in his original notation, where the dot means intersection,V is the universe, E is the membership relation, and so on.Jech (2003) uses the following set of 10 Gödel operations.

1. G1(X,Y ) = {X,Y }

2. G2(X,Y ) = X × Y

3. G3(X,Y ) = {(x, y) | x ∈ X, y ∈ Y, x ∈ y}

4. G4(X,Y ) = X − Y

5. G5(X,Y ) = X ∩ Y

6. G6(X) = ∪X

7. G7(X) = dom(X)

8. G8(X) = {(x, y) | (y, x) ∈ X}

9. G9(X) = {(x, y, z) | (x, z, y) ∈ X}

10. G10(X) = {(x, y, z) | (y, z, x) ∈ X}

29

30 CHAPTER 11. GÖDEL OPERATION

11.2 Properties

Gödel’s normal form theorem states that if φ(x1,...xn) is a formula with all quantifiers bounded, then the function{(x1,...,xn) ∈ X1×...×Xn | φ(x1, ..., xn)) of X1, ..., Xn is given by a composition of some Gödel operations.

11.3 References• Gödel, Kurt (1940). The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies 3. Prince-ton, N. J.: Princeton University Press. ISBN 978-0-691-07927-1. MR 0002514.

• Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-44085-7

Chapter 12

Inner model

In mathematical logic, suppose T is a theory in the language

L = ⟨∈⟩

of set theory.If M is a model of L describing a set theory and N is a class of M such that

⟨N,∈M , . . .⟩

is a model of T containing all ordinals of M then we say that N is an inner model of T (in M).[1] Ordinarily thesemodels are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension ofV.This term inner model is sometimes applied to models that are proper classes; the term set model is used for modelsthat are sets.A model of set theory is called standard if the element relation of the model is the actual element relation restrictedto the model. A model is called transitive when it is standard and the base class is a transitive class of sets. A modelof set theory is often assumed to be transitive unless it is explicitly stated that it is non-standard. Inner models aretransitive, transitive models are standard, and standard models are well-founded.The assumption that there exists a standard model of ZFC (in a given universe) is stronger than the assumptionthat there exists a model. In fact, if there is a standard model, then there is a smallest standard model called theminimal model contained in all standard models. The minimal model contains no standard model (as it is minimal)but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem. Thismodel is necessarily not well founded otherwise its Mostowski collapse would be a standard model. (It is not wellfounded as a relation in the universe, though it satisfies the axiom of foundation so is “internally” well founded. Beingwell founded is not an absolute property.[2]) In particular in the minimal model there is a model of ZFC but there isno standard model of ZFC.

12.1 Use

Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension ofZFC (like ZFC + ∃ a measurable cardinal). When no theory is mentioned, it is usually assumed that the model underdiscussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of subtheories of ZFC(like ZF or KP) as well.

31

32 CHAPTER 12. INNER MODEL

12.2 Related ideas

It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF (which is also an inner model ofZFC + GCH), called the constructible universe, or L.There is a branch of set theory called inner model theory that studies ways of constructing least inner models oftheories extending ZF. Inner model theory has led to the discovery of the exact consistency strength ofmany importantset theoretical properties.

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

[2] Kunen, Kenneth (1980). Set Theory. Amsterdam: North-Holland Pub. Co. ISBN 0-444-86839-9., Page 117

12.4 See also• Countable transitive models and generic filters

Chapter 13

Inner model theory

In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof.Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of ageneric extension of V. Inner model theory studies the relationships of these models to determinacy, large cardinals,and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model theory.

13.1 Examples• The class of all sets is an inner model containing all other inner models.

• The first non-trivial example of an inner model was the constructible universe L developed by Kurt Gödel.Every model M of ZFC has an inner model LM satisfying the axiom of constructibility, and this will be thesmallest inner model ofM containing all the ordinals ofM. Regardless of the properties of the original model,LM will satisfy the generalized continuum hypothesis and combinatorial axioms such as the diamond principle◊.

• The sets that are hereditarily ordinal definable form an inner model

• The sets that are hereditarily definable over a countable sequence of ordinals form an inner model, used inSolovay’s theorem.

• L(R)

• L[U] (see zero dagger)

13.2 Consistency results

One important use of inner models is the proof of consistency results. If it can be shown that every model of anaxiom A has an inner model satisfying axiom B, then if A is consistent, B must also be consistent. This analysis ismost useful when A is an axiom independent of ZFC, for example a large cardinal axiom; it is one of the tools usedto rank axioms by consistency strength.

13.3 References• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag

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

33

Chapter 14

Jech–Kunen tree

In mathematics, a Jech–Kunen tree is a tree of power ω1 and height ω1 in which the number of branches is greaterthan ω1 and less than 2ω1 . They are named after Thomas Jech (1971) who found the first example, and KennethKunen (1975) who related them to the cardinalities of compact spaces.

14.1 References• Jech, Thomas J. (1971), “Trees”, J. Symbolic Logic 36: 1–14, doi:10.2307/2271510, MR 0284331

• Kunen (1975), “On the cardinality of compact spaces”, Notices of the A. M. S. 22: 212

34

Chapter 15

Jensen hierarchy

In set theory, amathematical discipline, the Jensen hierarchy or J-hierarchy is amodification ofGödel's constructiblehierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchyfigures prominently in fine structure theory, a field pioneered by Ronald Jensen, for whom the Jensen hierarchy isnamed.

15.1 Definition

As in the definition of L, let Def(X) be the collection of sets definable with parameters over X:

Def(X) = { {y | y ε X and Φ(y, z1, ..., zn) is true in (X, ε)} | Φ is a first order formula and z1, ..., zn areelements of X}.

The constructible hierarchy, L is defined by transfinite recursion. In particular, at successor ordinals, Lα₊₁ = Def(Lα).The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; fora given x, y ε Lα₊₁ − Lα, the set {x,y} will not be an element of Lα₊₁, since it is not a subset of Lα.However, Lα does have the desirable property of being closed under Σ0 separation.Jensen’s modified hierarchy retains this property and the slightly weaker condition that Jα+1 ∩ Pow(Jα) = Def(Jα), but is also closed under pairing. The key technique is to encode hereditarily definable sets over Jα by codes; thenJα₊₁ will contain all sets whose codes are in Jα.Like Lα, Jα is defined recursively. For each ordinal α, we define Wα

n to be a universal Σ predicate for Jα. Weencode hereditarily definable sets asXα(n+ 1, e) = {X(n, f) | Wα

n+1(e, f)} , withXα(0, e) = e . Then set Jα,to be {X(n, e) | e in Jα}. Finally, Jα₊₁ =

∪n∈ω Jα,n .

15.2 Properties

Each sublevel Jα, n is transitive and contains all ordinals less than or equal to αω + n. The sequence of sublevels isstrictly increasing in n, since a Σm predicate is also Σn for any n > m. The levels Jα will thus be transitive and strictlyincreasing as well, and are also closed under pairing, Delta-0 comprehension and transitive closure. Moreover, theyhave the property that

Jα+1 ∩ Pow(Jα) = Def(Jα),

as desired.The levels and sublevels are themselves Σ1 uniformly definable [i.e. the definition of Jα, n in Jβ does not dependon β], and have a uniform Σ1 well-ordering. Finally, the levels of the Jensen hierarchy satisfy a condensation lemmamuch like the levels of Godel’s original hierarchy.

35

36 CHAPTER 15. JENSEN HIERARCHY

15.3 Rudimentary functions

A rudimentary function is a function that can be obtained from the following operations:

• F(x1, x2, ...) = xi is rudimentary

• F(x1, x2, ...) = {xi, xj} is rudimentary

• F(x1, x2, ...) = xi − xj is rudimentary

• Any composition of rudimentary functions is rudimentary

• ∪z∈yG(z, x1, x2, ...) is rudimentary

For any set M let rud(M) be the smallest set containing M∪{M} closed under the rudimentary operations. Then theJensen hierarchy satisfies Jα₊₁ = rud(Jα).

15.4 References• Sy Friedman (2000) Fine Structure and Class Forcing, Walter de Gruyter, ISBN 3-11-016777-8

Chapter 16

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.

16.1 See also• Paris–Harrington theorem

• Goodstein’s theorem

• Kruskal’s tree theorem

16.2 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

37

Chapter 17

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.

17.1 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.

17.2 See also

• Aronszajn tree

• Suslin tree

17.3 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.

38

17.3. REFERENCES 39

• 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 18

L(R)

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals andall the reals.

18.1 Construction

It can be constructed in a manner analogous to the construction of L (that is, Gödel’s constructible universe), byadding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.

18.2 Assumptions

In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannotshow even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiomof choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, givenonly that the von Neumann universe, V, also satisfies that axiom.

18.3 Results

Some additional results of the theory are:

• Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element ofL(R).

• Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property ofBaire and the perfect set property.

• L(R) does not satisfy the axiom of uniformization or the axiom of real determinacy.

• R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R).

• While not every relation on the reals in L(R) has a uniformization in L(R), every such relation does have auniformization in L(R#).

• Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated inV[G]. Thus the theory of L(R) cannot be changed by forcing.

• L(R) satisfies AD+.

40

18.4. REFERENCES 41

18.4 References• Woodin, W. Hugh (1988). “Supercompact cardinals, sets of reals, and weakly homogeneous trees”. Proceed-

ings of the National Academy of Sciences of theUnited States of America 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

Chapter 19

List of statements undecidable in ZFC

The mathematical statements discussed below are provably undecidable in ZFC (the Zermelo–Fraenkel axiomsplus the axiom of choice, the canonical axiomatic set theory of contemporary mathematics), assuming that ZFC isconsistent. A statement is undecidable in ZFC (a.k.a. independent of ZFC) if it can neither be proven nor disprovenfrom the axioms of ZFC.

19.1 Axiomatic set theory

In 1931, Kurt Gödel proved the first ZFC undecidability result, namely that the consistency of ZFC itself was unde-cidable in ZFC (Gödel’s second incompleteness theorem).Moreover, the following statements are undecidable in ZFC:

• The continuum hypothesis (CH); (Gödel produced a model of ZFC in which CH is true, showing that CHcannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC inwhich CH fails, showing that CH cannot be proven in ZFC. The following four undecidability results are alsodue to Gödel/Cohen.)

• The generalized continuum hypothesis (GCH);

• The axiom of constructibility (V = L);

• The diamond principle (◊);

• Martin’s axiom (MA);

• MA + ¬CH. (Undecidability shown by Solovay and Tennenbaum.)[1]

We have the following chains of implication:

V = L→ ◊ → CH.V = L→ GCH → CH.CH → MA

Another statement that is undecidable in ZFC is:

If the set S has fewer elements than T (in the sense of cardinality), then S also has fewer subsets than T.

Several statements related to the existence of large cardinals cannot be proven in ZFC (assuming ZFC is consistent).These are undecidable in ZFC provided that they are consistent with ZFC, which most working set theorists believeto be the case. These statements are strong enough to imply the consistency of ZFC. This has the consequence (viaGödel’s second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFCis consistent). The following statements belong to this class:

42

19.2. SET THEORY OF THE REAL LINE 43

• Existence of inaccessible cardinals

• Existence of Mahlo cardinals

• Existence of measurable cardinals (first conjectured by Ulam)

• Existence of supercompact cardinals

The following statements can be proven to be undecidable in ZFC assuming the consistency of a suitable large cardinal:

• Proper forcing axiom

• Open coloring axiom

• Martin’s maximum

• Existence of 0#

• Singular cardinals hypothesis

• Projective determinacy

19.2 Set theory of the real line

There are many cardinal invariants of the real line, connected with measure theory and statements related to the Bairecategory theorem, whose exact values are independent of ZFC.While nontrivial relations can be proved between them,most cardinal invariants can be any regular cardinal between ℵ1 and 2ℵ0 . This is a major area of study in the set theoryof the real line (see Cichon diagram). MA has a tendency to set most interesting cardinal invariants equal to 2ℵ0 .A subsetX of the real line is a strongmeasure zero set if to every sequence (εn) of positive reals there exists a sequenceof intervals (In) which covers X and such that In has length at most εn. Borel’s conjecture, that every strong measurezero set is countable, is undecidable in ZFC.A subset X of the real line is ℵ1 -dense if every open interval contains ℵ1 -many elements of X. Whether all ℵ1 -densesets are order-isomorphic is undecidable in ZFC.[2]

19.3 Order theory

Suslin’s problem asks whether a specific short list of properties characterizes the ordered set of real numbers R. Thisis undecidable in ZFC.[3] A Suslin line is an ordered set which satisfies this specific list of properties but is not order-isomorphic to R. The diamond principle ◊ proves the existence of a Suslin line, while MA + ¬CH implies EATS(every Aronszajn tree is special),[4] which in turn implies (but is not equivalent to)[5] the nonexistence of Suslin lines.Ronald Jensen proved that CH does not imply the existence of a Suslin line.[6]

Existence of Kurepa trees is undecidable in ZFC, assuming consistency of an inaccessible cardinal.[7]

Existence of a partition of the ordinal number ω2 into two colors with no monochromatic uncountable sequentiallyclosed subset is undecidable in ZFC, ZFC + CH, and ZFC + ¬CH, assuming consistency of a Mahlo cardinal.[8][9][10]This theorem of Shelah answers a question of H. Friedman.

19.4 Abstract algebra

In 1973, Saharon Shelah showed that the Whitehead problem (“is every abelian group A with Ext1(A, Z) = 0 a freeabelian group?") is undecidable in ZFC.[11] An abelian group with Ext1(A, Z) = 0 is called a Whitehead group; MA+ ¬CH proves the existence of a non-free Whitehead group, while V = L proves that all Whitehead groups are free.In one of the earliest applications of proper forcing, Shelah constructed a model of ZFC + CH in which there is anon-free Whitehead group.[12][13]

44 CHAPTER 19. LIST OF STATEMENTS UNDECIDABLE IN ZFC

Consider the ring A=R[x,y,z] of polynomials in three variables over the real numbers and its field of fractionsM=R(x,y,z). The projective dimension of M as A-module is either 2 or 3, but it is undecidable in ZFC whetherit is equal to 2; it is equal to 2 if and only if CH holds.[14]

A direct product of countably many fields has global dimension 2 if and only if the continuum hypothesis holds.[15]

19.5 Number theory

One can write down a concrete polynomial P∈Z[x1,...x9] such the statement “there are integers m1,...,m9 withP(m1,...,m9)=0” can neither be proven nor disproven in ZFC (assuming ZFC is consistent).[16] This follows fromYuri Matiyasevich's resolution of Hilbert’s tenth problem; the polynomial is constructed so that it has an integer rootif and only if ZFC is inconsistent.

19.6 Measure theory

A stronger version of Fubini’s theorem for positive functions, where the function is no longer assumed to bemeasurablebut merely that the two iterated integrals are well defined and exist, is undecidable in ZFC. On the one hand, CHimplies that there exists a function on the unit square whose iterated integrals are not equal — the function is simplythe indicator function of an ordering of [0, 1] equivalent to a well ordering of the cardinal ω1. A similar examplecan be constructed using MA. On the other hand, the consistency of the strong Fubini theorem was first shown byFriedman.[17] It can also be deduced from a variant of Freiling’s axiom of symmetry.[18]

19.7 Topology

TheNormalMoore Space conjecture, namely that every normalMoore space ismetrizable, can be disproven assumingCH or MA + ¬CH, and can be proven assuming a certain axiom which implies the existence of large cardinals. Sincethe existence of large cardinals has not been proven to be consistent with ZFC, we cannot yet say that the NormalMoore Space conjecture is undecidable in ZFC.Various assertions about P (ω)/ finite, P-points, Q-points,...S- and L- spaces

19.8 Functional analysis

Garth Dales and Robert M. Solovay proved in 1976 that Kaplansky’s conjecture, namely that every algebra homomor-phism from the Banach algebra C(X) (where X is some compact Hausdorff space) into any other Banach algebra mustbe continuous, is independent of ZFC. CH implies that for any infinite X there exists a discontinuous homomorphisminto any Banach algebra.[19]

Consider the algebra B(H) of bounded linear operators on the infinite-dimensional separable Hilbert space H. Thecompact operators form a two-sided ideal in B(H). The question of whether this ideal is the sum of two properlysmaller ideals is undecidable in ZFC, as was proved by Andreas Blass and Saharon Shelah in 1987.[20]

Charles Akemann and Nik Weaver showed in 2003 that the statement “there exists a counterexample to Naimark’sproblem which is generated by ℵ1, elements” is independent of ZFC.Miroslav Bačák and Petr Hájek proved in 2008 that the statement “every Asplund space of density character ω1

has a renorming with the Mazur intersection property" is independent of ZFC. The result is shown using Martin’smaximum axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample assuming CH.As shown by Ilijas Farah[21] and N. Christopher Phillips and Nik Weaver,[22] the existence of outer automorphismsof the Calkin algebra depends on set theoretic assumptions beyond ZFC.

19.9. MODEL THEORY 45

19.9 Model theory

Chang’s conjecture is undecidable assuming the consistency of an Erdős cardinal.

19.10 References[1] Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

[2] Baumgartner, J., All ℵ1 -dense sets of reals can be isomorphic, Fund. Math. 79, pp.101 -- 106, 1973

[3] Solovay, R. M.; Tennenbaum, S. (1971). “Iterated Cohen extensions and Souslin’s problem”. Annals Of Mathematics.Second Series 94 (2): 201–245. doi:10.2307/1970860. JSTOR 1970860.

[4] Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Nat. Acad. Science, U.S.A., 67, pp.1746 -- 1753, 1970

[5] Shelah, S., Free limits of forcing and more on Aronszajn trees, Israel Journal of Mathematics, 40, pp. 1 -- 32, 1971

[6] Devlin, K., and H. Johnsbraten, The Souslin Problem, Lecture Notes on Mathematics 405, Springer, 1974

[7] Silver, J., The independence of Kurepa’s conjecture and two-cardinal conjectures inmodel theory, in Axiomatic Set Theory,Proc. Symp, in Pure Mathematics (13) pp. 383 - 390, 1967

[8] Shelah, S., Proper and Improper Forcing, Springer 1992

[9] Schlindwein, Chaz, Shelah’s work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 --606

[10] Schlindwein, Chaz, Shelah’s work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865 -- 1883

[11] Shelah, S. (1974). “Infinite Abelian groups, Whitehead problem and some constructions”. Israel Journal of Mathematics18: 243–256. doi:10.1007/BF02757281. MR 0357114.

[12] Shelah, S., Whitehead groups may not be free even assuming CH I, Israel Journal of Mathematics (28) 1972

[13] Shelah, S., Whitehead groups may not be free even assuming CH II, Israel Journal of Mathematics (350 1980

[14] Barbara L. Osofsky (1968). “Homological dimension and the continuum hypothesis” (PDF). Transactions of the AmericanMathematical Society: 217–230.

[15] Barbara L. Osofsky (1973). Homological Dimensions of Modules. American Mathematical Soc. p. 60.

[16] James P. Jones (1980). “Undecidable diophantine equations”. Bull. Amer. Math. Soc. 3 (2): 859–862. doi:10.1090/s0273-0979-1980-14832-6.

[17] Friedman, Harvey (1980). “A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions”. Illinois J. Math. 24 (3):390–395. MR 573474.

[18] Freiling, Chris (1986). “Axioms of symmetry: throwing darts at the real number line”. Journal of Symbolic Logic 51 (1):190–200. doi:10.2307/2273955. JSTOR 2273955. MR 830085.

[19] H. G. Dales, W. H. Woodin (1987). An introduction to independence for analysts.

[20] Judith Roitman (1992). “The Uses of Set Theory”. Mathematical Intelligencer 14 (1).

[21] Farah, Ilijas (2007). “All automorphisms of the Calkin algebra are inner”. arXiv:0705.3085.

[22] Phillips, N. C.; Weaver, N. (2007). “The Calkin algebra has outer automorphisms”. Duke Mathematical Journal 139 (1):185–202. doi:10.1215/S0012-7094-07-13915-2.

19.11 External links• What are some reasonable-sounding statements that are independent of ZFC?, mathoverflow.net

46 CHAPTER 19. LIST OF STATEMENTS UNDECIDABLE IN ZFC

Diagram showing the implication chains

Chapter 20

Minimal model (set theory)

In set theory, a minimal model is a minimal standard model of ZFC. Minimal models were introduced by (Shep-herdson 1951, 1952, 1953).The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows fromthe existence of a standard model as follows. If there is a set W in the von Neumann universe V which is a standardmodel of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the class of constructible sets ofW. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. This set is called theminimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolemtheorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s ofthe minimal model can be named; in other words there is a first order sentence φ(x) such that s is the unique elementof the minimal model for which φ(s) is true.Cohen (1963) gave another construction of the minimal model as the strongly constructible sets, using a modifiedform of Godel’s constructible universe.Of course, any consistent theory must have a model, so even within the minimal model of set theory there are setswhich are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, theydo not use the normal element relation and they are not well founded.If there is no standard model then the minimal model cannot exist as a set. However in this case the class of allconstructible sets plays the same role as the minimal model and has similar properties (though it is now a proper classrather than a countable set).

20.1 References• Cohen, Paul J. (1963), “Aminimalmodel for set theory”, Bull. Amer. Math. Soc. 69: 537–540, doi:10.1090/S0002-9904-1963-10989-1, MR 0150036

• Shepherdson, J. C. (1951), “Inner models for set theory. I”, The Journal of Symbolic Logic (Association forSymbolic Logic) 16 (3): 161–190, doi:10.2307/2266389, JSTOR 2266389, MR 0045073

• Shepherdson, J. C. (1952), “Inner models for set theory. II”, The Journal of Symbolic Logic (Association forSymbolic Logic) 17 (4): 225–237, doi:10.2307/2266609, JSTOR 2266609, MR 0053885

• Shepherdson, J. C. (1953), “Inner models for set theory. III”, The Journal of Symbolic Logic (Association forSymbolic Logic) 18 (2): 145–167, doi:10.2307/2268947, JSTOR 2268947, MR 0057828

47

Chapter 21

Mouse (set theory)

In set theory, a mouse is a small model of (a fragment of) Zermelo–Fraenkel set theory with desirable properties.The exact definition depends on the context. In most cases, there is a technical definition of “premouse” and an addedcondition of iterability (referring to the existence of wellfounded iterated ultrapowers): a mouse is then an iterablepremouse. The notion of mouse generalizes the concept of a level of Gödel's constructible hierarchy while being ableto incorporate large cardinals.Mice are important ingredients of the construction of core models. The concept was isolated by Ronald Jensen in the1970s and has been used since then in core model constructions of many authors. An urban legend says that “mice”was originally a misprint for “nice”, but Jensen has denied this.

21.1 References• Dodd, A.; Jensen, R. (1981), “The core model”, Ann. Math. Logic 20 (1): 43–75, MR 0611394

• 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.

• Mitchell, William (1979), “Ramsey cardinals and constructibility”, J. Symbolic Logic 44 (2): 260–266, MR0534574

48

Chapter 22

Naimark’s problem

Naimark's Problem is a question in functional analysis. It asks whether every C*-algebra that has only one irre-ducible ∗ -representation up to unitary equivalence is isomorphic to the ∗ -algebra of compact operators on some (notnecessarily separable) Hilbert space.The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras).Akemann &Weaver (2004) used the♢ -Principle to construct a C*-algebra with ℵ1 generators that serves as a coun-terexample to Naimark’s Problem. More precisely, they showed that the statement "There exists a counterexampleto Naimark’s Problem that is generated by ℵ1 elements" is independent of the axioms of Zermelo-Fraenkel SetTheory and the Axiom of Choice ( ZFC ).Whether Naimark’s problem itself is independent of ZFC remains unknown.

22.1 See also• List of statements undecidable in ZFC

• Gelfand-Naimark Theorem

22.2 External links• Akemann, Charles; Weaver, Nik (2004), “Consistency of a counterexample toNaimark’s problem”, Proceedings

of the National Academy of Sciences of the United States of America 101 (20): 7522–7525, arXiv:math.OA/0312135, doi:10.1073/pnas.0401489101, MR 2057719

49

Chapter 23

Paris–Harrington theorem

Inmathematical logic, theParis–Harrington theorem states that a certain combinatorial principle in Ramsey theory,namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic. This was the first“natural” example of a true statement about the integers that could be stated in the language of arithmetic, but notproved in Peano arithmetic; it was already known that such statements existed byGödel’s first incompleteness theorem.

23.1 The strengthened finite Ramsey theorem

The strengthened finite Ramsey theorem is a statement about colorings and natural numbers and states that:

• For any positive integers n, k, m we can find N with the following property: if we color each of the n-elementsubsets of S = {1, 2, 3,..., N} with one of k colors, then we can find a subset Y of S with at least m elements,such that all n element subsets of Y have the same color, and the number of elements of Y is at least the smallestelement of Y.

Without the condition that the number of elements of Y is at least the smallest element of Y, this is a corollary of thefinite Ramsey theorem inKPn(S) , with N given by:

(N

n

)= |Pn(S)| ≥ R(m,m, . . . ,m︸ ︷︷ ︸

k

).

Moreover the strengthened finite Ramsey theorem can be deduced from the infinite Ramsey theorem in almost exactlythe same way that the finite Ramsey theorem can be deduced from it, using a compactness argument (see the articleon Ramsey’s theorem for details). This proof can be carried out in second-order arithmetic.The Paris–Harrington theorem states that the strengthened finite Ramsey theorem is not provable in Peano arithmetic.

23.2 The Paris–Harrington theorem

Roughly speaking, Jeff Paris and Leo Harrington showed that the strengthened finite Ramsey theorem is unprovablein Peano arithmetic by showing that in Peano arithmetic it implies the consistency of Peano arithmetic itself. SincePeano arithmetic cannot prove its own consistency by Gödel’s theorem, this shows that Peano arithmetic cannot provethe strengthened finite Ramsey theorem.The smallest number N that satisfies the strengthened finite Ramsey theorem is a computable function of n, m, k, butgrows extremely fast. In particular it is not primitive recursive, but it is also far larger than standard examples of nonprimitive recursive functions such as the Ackermann function. Its growth is so large that Peano arithmetic cannotprove it is defined everywhere, although Peano arithmetic easily proves that the Ackermann function is well defined.

50

23.3. SEE ALSO 51

23.3 See also• Kanamori–McAloon theorem

• Goodstein’s theorem

• Kruskal’s tree theorem

23.4 References• David Marker, Model Theory: An Introduction, ISBN 0-387-98760-6

• mathworld entry

• Paris, J. and Harrington, L.AMathematical Incompleteness in Peano Arithmetic. In Handbook ofMathematicalLogic (Ed. J. Barwise). Amsterdam, Netherlands: North-Holland, 1977.

23.5 External links• A brief introduction to unprovability (contains a proof of the Paris–Harrington theorem) by Andrey Bovykin.

Chapter 24

Silver machine

This article is about the kind of mathematical object. For the Hawkwind song, see Silver Machine. For the Vaporssong, see Silver Machines.

In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holdingin L. They were invented by set theorist Jack Silver as a means of proving global square holds in the constructibleuniverse.

24.1 Preliminaries

An ordinal α is *definable from a class of ordinals X if and only if there is a formula ϕ(µ0, µ1, . . . , µn) and∃β1, . . . , βn, γ ∈ X such that α is the unique ordinal for which |=Lγ

ϕ(α◦, β◦1 , . . . , β

◦n) where for all α we de-

fine α◦ to be the name for α within Lγ .A structure ⟨X,<, (hi)i<ω⟩ is eligible if and only if:

1. X ⊆ On .

2. < is the ordering on On restricted to X.

3. ∀i, hi is a partial function from Xk(i) to X, for some integer k(i).

If N = ⟨X,<, (hi)i<ω⟩ is an eligible structure then Nλ is defined to be as before but with all occurrences of Xreplaced withX ∩ λ .Let N1, N2 be two eligible structures which have the same function k. Then we say N1 ◁ N2 if ∀i ∈ ω and∀x1, . . . , xk(i) ∈ X1 we have:h1i (x1, . . . , xk(i)) ∼= h2

i (x1, . . . , xk(i))

24.2 Silver machine

A Silver machine is an eligible structure of the formM = ⟨On,<, (hi)i<ω⟩ which satisfies the following conditions:Condensation principle. If N ◁Mλ then there is an α such that N ∼= Mα .Finiteness principle. For each λ there is a finite setH ⊆ λ such that for any set A ⊆ λ+ 1 we have

Mλ+1[A] ⊆ Mλ[(A ∩ λ) ∪H] ∪ {λ}

Skolem property. If α is *definable from the set X ⊆ On , then α ∈ M [X] ; moreover there is an ordinal λ <[sup(X) ∪ α]+ , uniformly Σ1 definable from X ∪ {α} , such that α ∈ Mλ[X] .

52

24.3. REFERENCES 53

24.3 References• Keith J Devlin (1984). “Chapter IX”. Constructibility. ISBN 0-387-13258-9. - Please note that errors have beenfound in some results in this book concerning Kripke Platek set theory.

Chapter 25

Skolem’s paradox

In mathematical logic and philosophy, Skolem’s paradox is a seeming contradiction that arises from the downwardLöwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects ofthe theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is notan actual antinomy like Russell’s paradox, the result is typically called a paradox, and was described as a “paradoxicalstate of affairs” by Skolem (1922: p. 295).Skolem’s paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a modelthat is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence thatintuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable.Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countablesets, satisfies the first order sentence that intuitively states “there are uncountable sets”.Amathematical explanation of the paradox, showing that it is not a contradiction inmathematics, was given by Skolem(1922). Skolem’s work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic,but the result quickly came to be accepted by the mathematical community.The philosophical implications of Skolem’s paradox have received much study. One line of inquiry questions whetherit is accurate to claim that any first-order sentence actually states “there are uncountable sets”. This line of thoughtcan be extended to question whether any set is uncountable in an absolute sense. More recently, the paper “Modelsand Reality” by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of Skolem’sresult.

25.1 Background

One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets,such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set X is countableif there is a function that gives a one-to-one correspondence between X and the natural numbers, and is uncountableif there is no such correspondence function. When Zermelo proposed his axioms for set theory in 1908, he provedCantor’s theorem from them to demonstrate their strength.Löwenheim (1915) and Skolem (1920, 1923) proved the Löwenheim–Skolem theorem. The downward form of thistheorem shows that if a countable first-order axiomatisation is satisfied by any infinite structure, then the same axiomsare satisfied by some countable structure. In particular, this implies that if the first order versions of Zermelo’s axiomsof set theory are satisfiable, they are satisfiable in some countable model. The same is true of any consistent first orderaxiomatisation of set theory.

25.2 The paradoxical result and its mathematical implications

Skolem (1922) pointed out the seeming contradiction between the Löwenheim–Skolem theorem on the one hand,which implies that there is a countable model of Zermelo’s axioms, and Cantor’s theorem on the other hand, whichstates that uncountable sets exist, and which is provable from Zermelo’s axioms. “So far as I know,” Skolem writes,

54

25.3. RECEPTION BY THE MATHEMATICAL COMMUNITY 55

“no one has called attention to this peculiar and apparently paradoxical state of affairs. By virtue of the axioms wecan prove the existence of higher cardinalities... How can it be, then, that the entire domain B [a countable modelof Zermelo’s axioms] can already be enumerated by means of the finite positive integers?" (Skolem 1922, p. 295,translation by Bauer-Mengelberg)More specifically, let B be a countable model of Zermelo’s axioms. Then there is some set u in B such that B satisfiesthe first-order formula saying that u is uncountable. For example, u could be taken as the set of real numbers in B.Now, because B is countable, there are only countably many elements c such that c ∈ u according to B, because thereare only countably many elements c in B to begin with. Thus it appears that u should be countable. This is Skolem’sparadox.Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term“set” does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition ofcountability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible torecognize that a particular set u is countable, but not countable in a particular model of set theory, because there isno set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.Skolem used the term “relative” to describe this state of affairs, where the same set is included in two models of settheory, is countable in one model, and is not countable in the other model. He described this as the “most important”result in his paper. Contemporary set theorists describe concepts that do not depend on the choice of a transitivemodel as absolute. From their point of view, Skolem’s paradox simply shows that countability is not an absoluteproperty in first order logic. (Kunen 1980 p. 141; Enderton 2001 p. 152; Burgess 1977 p. 406).Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a foundationalsystem:

“I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foun-dation of mathematics that mathematicians would, for the most part, not be very much concerned withit. But in recent times I have seen to my surprise that so many mathematicians think that these axiomsof set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time hadcome for a critique.” (Ebbinghaus and van Dalen, 2000, p. 147)

25.3 Reception by the mathematical community

A central goal of early research into set theory was to find a first order axiomatisation for set theory which wascategorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem’s result showedthis is not possible, creating doubts about the use of set theory as a foundation of mathematics. It took some time forthe theory of first-order logic to be developed enough for mathematicians to understand the cause of Skolem’s result;no resolution of the paradox was widely accepted during the 1920s. Fraenkel (1928) still described the result as anantinomy:

“Neither have the books yet been closed on the antinomy, nor has agreement on its significance andpossible solution yet been reached.” (van Dalen and Ebbinghaus, 2000, p. 147).

In 1925, von Neumann presented a novel axiomatization of set theory, which developed into NBG set theory. Verymuch aware of Skolem’s 1922 paper, von Neumann investigated countable models of his axioms in detail. In hisconcluding remarks, Von Neumann comments that there is no categorical axiomatization of set theory, or any othertheory with an infinite model. Speaking of the impact of Skolem’s paradox, he wrote,

“At present we can do no more than note that we have one more reason here to entertain reservationsabout set theory and that for the time being no way of rehabilitating this theory is known."(Ebbinghausand van Dalen, 2000, p. 148)

Zermelo at first considered the Skolem paradox a hoax (van Dalen and Ebbinghaus, 2000, p. 148 ff.), and spokeagainst it starting in 1929. Skolem’s result applies only to what is now called first-order logic, but Zermelo arguedagainst the finitary metamathematics that underlie first-order logic (Kanamori 2004, p. 519 ff.). Zermelo argued thathis axioms should instead be studied in second-order logic, a setting in which Skolem’s result does not apply. Zermelopublished a second-order axiomatization in 1930 and proved several categoricity results in that context. Zermelo’s

56 CHAPTER 25. SKOLEM’S PARADOX

further work on the foundations of set theory after Skolem’s paper led to his discovery of the cumulative hierarchyand formalization of infinitary logic (van Dalen and Ebbinghaus, 2000, note 11).Fraenkel et al. (1973, pp. 303–304) explain why Skolem’s result was so surprising to set theorists in the 1920s.Gödel’s completeness theorem and the compactness theorem were not proved until 1929. These theorems illumi-nated the way that first-order logic behaves and established its finitary nature, although Gödel’s original proof of thecompleteness theorem was complicated. Leon Henkin’s alternative proof of the completeness theorem, which is nowa standard technique for constructing countable models of a consistent first-order theory, was not presented until1947. Thus, in 1922, the particular properties of first-order logic that permit Skolem’s paradox to go through werenot yet understood. It is now known that Skolem’s paradox is unique to first-order logic; if set theory is studied usinghigher-order logic with full semantics then it does not have any countable models, due to the semantics being used.

25.4 Current mathematical opinion

Current mathematical logicians do not view Skolem’s paradox as any sort of fatal flaw in set theory. Kleene (1967,p. 324) describes the result as “not a paradox in the sense of outright contradiction, but rather a kind of anomaly”.After surveying Skolem’s argument that the result is not contradictory, Kleene concludes “there is no absolute notionof countability.” Hunter (1971, p. 208) describes the contradiction as “hardly even a paradox”. Fraenkel et al. (1973,p. 304) explain that contemporary mathematicians are no more bothered by the lack of categoricity of first-ordertheories than they are bothered by the conclusion of Gödel’s incompleteness theorem that no consistent, effective, andsufficiently strong set of first-order axioms is complete.Countable models of ZF have become common tools in the study of set theory. Forcing, for example, is oftenexplained in terms of countable models. The fact that these countable models of ZF still satisfy the theorem that thereare uncountable sets is not considered a pathology; van Heijenoort (1967) describes it as “a novel and unexpectedfeature of formal systems.” (van Heijenoort 1967, p. 290)Althoughmathematicians no longer consider Skolem’s result paradoxical, the result is often discussed by philosophers.In the setting of philosophy, a merely mathematical resolution of the paradox may be less than satisfactory.

25.5 References

• Barwise, Jon (1977), “An introduction to first-order logic”, in Barwise, Jon, ed. (1982), Handbook of Math-ematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN978-0-444-86388-1

• Timothy Bays (2000). Reflections on Skolem’s Paradox (PDF) (Ph.D. thesis). UCLA Philosophy Department.

• Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972), What is mathematical logic?,London-Oxford-New York: Oxford University Press, ISBN 0-19-888087-1, Zbl 0251.02001

• Dirk Van Dalen; Heinz-Dieter Ebbinghaus (Jun 2000). “Zermelo and the Skolem Paradox”. The Bulletin ofSymbolic Logic 6 (2): 145—161.

• Dragalin, A.G. (2001), “S/s085750”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Abraham Fraenkel, Yehoshua Bar-Hillel, Azriel Levy, Dirk van Dalen (1973), Foundations of Set Theory,North-Holland.

• Henkin, L. (1950), “Completeness in the theory of types”, The Journal of Symbolic Logic 15 (2): 81–91,doi:10.2307/2266967, JSTOR 2266967.

• Kanamori, Akihiro (2004), “Zermelo and set theory”, The Bulletin of Symbolic Logic 10 (4): 487–553, doi:10.2178/bsl/1102083759,ISSN 1079-8986, MR 2136635

• Stephen Cole Kleene, (1952, 1971 with emendations, 1991 10th printing), Introduction to Metamathematics,North-Holland Publishing Company, Amsterdam NY. ISBN 0-444-10088-1. cf pages 420-432: § 75. Axiomsystems, Skolem’s paradox, the natural number sequence.

25.6. EXTERNAL LINKS 57

• Stephen Cole Kleene, (1967). Mathematical Logic.

• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North-Holland,ISBN 978-0-444-85401-8

• Löwenheim, Leopold (1915), "Über Möglichkeiten im Relativkalkül” (PDF), Mathematische Annalen 76 (4):447–470, doi:10.1007/BF01458217, ISSN 0025-5831

• Moore, A.W., “Set Theory, Skolem’s Paradox and the Tractatus”, Analysis 1985: 45, doi:10.2307/3327397.

• Hilary Putnam (Sep 1980). “Models and Reality” (PDF). The Journal of Symbolic Logic 45 (3): 464—482.

• Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: SpringerScience+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

• Skolem, Thoralf (1922). “Axiomatized set theory”. Reprinted in From Frege to Gödel, van Heijenoort, 1967,in English translation by Stefan Bauer-Mengelberg, pp. 291–301.

25.6 External links• Vaughan Pratt’s celebration of his academic ancestor Skolem’s 120th birthday

• Extract from Moore’s discussion of the paradox(broken link)

Chapter 26

Statements true in L

Here is a list of propositions that hold in the constructible universe (denoted L):

• The generalized continuum hypothesis and as a consequence

• The axiom of choice

• Diamondsuit

• Clubsuit

• Global square

• The existence of morasses

• The negation of the Suslin hypothesis

• The non-existence of 0# and as a consequence

• The non existence of all large cardinals which imply the existence of a measurable cardinal

• The truth of Whitehead’s conjecture that every abelian group A with Ext1(A, Z) = 0 is a free abelian group.

• The existence of a definable well-order of all sets (the formula for which can be given explicitly). In particular,L satisfies V=HOD.

Accepting the axiom of constructibility (which asserts that every set is constructible) these propositions also hold inthe von Neumann universe, resolving many propositions in set theory and some interesting questions in analysis.

58

Chapter 27

Strong measure zero set

In mathematical analysis, a strong measure zero set[1] is a subset A of the real line with the following property:

for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn forall n and A is contained in the union of the In.

(Here |In| denotes the length of the interval In.)Every countable set is a strong measure set, and so is every union of countably many strong measure zero sets. Everystrong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesguemeasure 0 which is not of strong measure zero.[2]

Borel’s conjecture[1] states that every strong measure zero set is countable. It is now known that this statement isindependent of ZFC (the Zermelo–Fraenkel axioms of set theory, which is the standard axiom system assumed inmathematics). This means that Borel’s conjecture can neither be proven nor disproven in ZFC (assuming ZFC isconsistent). Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent ofZFC) implies the existence of uncountable strong measure zero sets.[3] In 1976 Laver used a method of forcing toconstruct a model of ZFC in which Borel’s conjecture holds.[4] These two results together establish the independenceof Borel’s conjecture.It is known that if A ⊆ R has Lebesgue measure zero and M ⊆ R is a meagre set, then A + M ≠ R. The followingcharacterization of strong measure zero sets was proved in 1973:

A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R.[5]

This result establishes a connection to the notion of strongly meagre set, defined as follows:

A setM ⊆ R is strongly meagre if and only if A +M ≠ R for every set A ⊆ R of Lebesgue measure zero.

The dual Borel conjecture states that every strongly meagre set is countable. This statement is also independent ofZFC.[6]

27.1 References[1] E. Borel, Sur la classification des ensembles de mesure nulle, Bull. Soc. Math. France 47 (1919), 97–125.

[2] Thomas Jech (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer Science & BusinessMedia.

[3] W. Sierpiński, “Sur un ensemble non denombrable, dont toute image continue est de mesure nulle”, Fundamenta Mathe-maticae 11 (1928), 302–304

[4] R. Laver: On the consistency of Borel’s conjecture, Acta Math., 137(1976), 151–169.

59

60 CHAPTER 27. STRONG MEASURE ZERO SET

[5] Galvin, F., Mycielski, J., & Solovay, R.M. (1973). Strong measure zero sets. Notices of the AmericanMathematical Society,26.

[6] Timothy J. Carlson. Strong measure zero and strongly meager sets. Proc. Amer. Math. Soc., 118(2):577–586, 1993

Chapter 28

Suslin’s problem

In mathematics, Suslin’s problem is a question about totally ordered sets posed posthumously by Mikhail Yakovle-vich Suslin (1920). It has been shown to be independent of the standard axiomatic system of set theory known asZFC: the statement can neither be proven nor disproven from those axioms.[1]

(Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)Un ensemble ordonné (linéairement) sans sauts ni lacunes et tel que tout ensemble de ses intervalles (contenant plusqu'un élément) n'empiétant pas les uns sur les autres est au plus dénumerable, est-il nécessairement un continuelinéaire (ordinaire)?A (linearly) ordered set without jumps or gaps and such that every set of its intervals (containing more than oneelement) not overlapping each other is at most denumerable, is it necessarily an (ordinary) linear continuum?The original statement of Suslin’s problem from (Suslin 1920)

28.1 Formulation

Given a non-empty totally ordered set R with the following four properties:

1. R does not have a least nor a greatest element;

2. the order on R is dense (between any two elements there is another);

3. the order on R is complete, in the sense that every non-empty bounded subset has a supremum and an infimum;

4. every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chaincondition for the order topology of R).

Is R necessarily order-isomorphic to the real line R?If the requirement for the countable chain condition is replaced with the requirement that R contains a countabledense subset (i.e., R is a separable space) then the answer is indeed yes: any such set R is necessarily isomorphic toR (proved by Cantor).The condition for a topological space that every collection of non-empty disjoint open sets is at most countable iscalled the Suslin property.

28.2 Implications

Any totally ordered set that is not isomorphic to R but satisfies (1) – (4) is known as a Suslin line. The Suslinhypothesis says that there are no Suslin lines: that every countable-chain-condition dense complete linear orderwithout endpoints is isomorphic to the real line. Equivalently, that every tree of height ω1 either has a branch of

61

62 CHAPTER 28. SUSLIN’S PROBLEM

length ω1 or an antichain of cardinality ℵ1. The generalized Suslin hypothesis says that for every infinite regularcardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ. The existence ofSuslin lines is equivalent to the existence of Suslin trees and to Suslin algebras.The Suslin hypothesis is independent of ZFC. Jech (1967) and Tennenbaum (1968) independently used forcing meth-ods to construct models of ZFC in which Suslin lines exist. Jensen later proved that Suslin lines exist if the diamondprinciple, a consequence of the Axiom of constructibility V=L, is assumed. (Jensen’s result was a surprise as ithad previously been conjectured that V=L implies that no Suslin lines exist, on the grounds that V=L implies thereare “few” sets.) On the other hand, Solovay & Tennenbaum (1971) used forcing to construct a model of ZFC inwhich there are no Suslin lines; more precisely they showed that Martin’s axiom plus the negation of the ContinuumHypothesis implies the Suslin Hypothesis.The Suslin hypothesis is also independent of both the generalized continuum hypothesis (proved by Ronald Jensen)and of the negation of the continuum hypothesis. It is not known whether the Generalized Suslin Hypothesis isconsistent with the Generalized Continuum Hypothesis; however, since the combination implies the negation ofthe square principle at a singular strong limit cardinal—in fact, at all singular cardinals and all regular successorcardinals—it implies that the axiom of determinacy holds in L(R) and is believed to imply the existence of an innermodel with a superstrong cardinal.

28.3 See also• List of statements undecidable in ZFC

• AD+

28.4 Notes[1] Solovay, R. M.; Tennenbaum, S. (1971). “Iterated Cohen extensions and Souslin’s problem”. Ann. Of Math. (2) (Annals

of Mathematics) 94 (2): 201–245. doi:10.2307/1970860. JSTOR 1970860.

• K. Devlin and H. Johnsbråten, The Souslin Problem, Lecture Notes in Mathematics (405) Springer 1974

• Jech, Tomáš (1967), “Non-provability of Souslin’s hypothesis”, Comment. Math. Univ. Carolinae 8: 291–305,MR 0215729

• Souslin, M. (1920), “Problème 3” (PDF), Fundamenta Mathematicae 1: 223

• Tennenbaum, S. (1968), “Souslin’s problem.”, Proc. Nat. Acad. Sci. U.S.A. 59: 60–63, doi:10.1073/pnas.59.1.60,MR 0224456

28.5 References• Grishin, V.N. (2001), “Suslin hypothesis”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Chapter 29

Wetzel’s problem

In mathematics, Wetzel’s problem concerns bounds on the cardinality of a set of analytic functions that, for eachof their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University ofIllinois at Urbana–Champaign.[1][2]

Let F be a family of distinct analytic functions on a given domain with the property that, for each x in the domain, thefunctions in F map x to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumptionimplies that F is necessarily itself countable.[3] Paul Erdős in turn learned about the problem at the University ofMichigan, likely via Lee Albert Rubel.[1] In his paper on the problem, Erdős credited an anonymous mathematicianwith the observation that, when each x is mapped to a finite set of values, F is necessarily finite.[4]

However, as Erdős showed, the situation for countable sets is more complicated: the answer toWetzel’s question is yesif and only if the continuum hypothesis is false.[4] That is, the existence of an uncountable set of functions that mapsany argument x to a countable set of values is equivalent to the nonexistence of an uncountable set of real numberswhose cardinality is less than the cardinality of the set of all real numbers. One direction of this equivalence was alsoproven independently, but not published, by another UIUC mathematician, Robert Dan Dixon.[1] It follows from theindependence of the continuum hypothesis, proved in 1963 by Paul Cohen,[5] that the answer to Wetzel’s problem isindependent of ZFC set theory.[1]

29.1 References[1] Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015), “Wetzel’s problem, Paul Erdős, and the continuum hypothesis:

a mathematical mystery”, Notices of the AMS 62 (3): 243–247, arXiv:1406.5085.

[2] Aigner, Martin; Ziegler, Günter M. (2014), Proofs from The Book (5th ed.), Springer-Verlag, Berlin, pp. 132–134,doi:10.1007/978-3-662-44205-0, ISBN 978-3-662-44204-3, MR 3288091.

[3] Wetzel, John Edward (1964), A Compactification Theory with Potential-Theoretic Applications, Ph.D. thesis, Stanford Uni-versity, p. 98. As cited by Garcia & Shoemaker (2015).

[4] Erdős, P. (1964), “An interpolation problem associated with the continuum hypothesis”, The Michigan Mathematical Jour-nal 11: 9–10, MR 0168482.

[5] Cohen, Paul J. (December 15, 1963), “The Independence of the Continuum Hypothesis”, Proceedings of the NationalAcademy of Sciences of the United States of America 50 (6): 1143–1148, doi:10.1073/pnas.50.6.1143, JSTOR 71858,PMID 16578557.

63

Chapter 30

Whitehead problem

In group theory, a branch of abstract algebra, theWhitehead problem is the following question:

Is every abelian group A with Ext1(A, Z) = 0 a free abelian group?

Shelah (1974) proved that Whitehead’s problem is undecidable within standard ZFC set theory.

30.1 Refinement

The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B→ A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there existsa group homomorphism g : A → B with fg = idA. Abelian groups A satisfying this condition are sometimes calledWhitehead groups, so Whitehead’s problem asks: is every Whitehead group free?Caution: The converse of Whitehead’s problem, namely that every free abelian group is Whitehead, is a well knowngroup-theoretical fact. Some authors callWhitehead group only a non-free group A satisfying Ext1(A, Z) = 0. White-head’s problem then asks: do Whitehead groups exist?

30.2 Shelah’s proof

Saharon Shelah (1974) showed that, given the canonical ZFC axiom system, the problem is independent of the usualaxioms of set theory. More precisely, he showed that:

• If every set is constructible, then every Whitehead group is free;• If Martin’s axiom and the negation of the continuum hypothesis both hold, then there is a non-free Whiteheadgroup.

Since the consistency of ZFC implies the consistency of either of the following:

• The axiom of constructibility (which asserts that all sets are constructible);• Martin’s axiom plus the negation of the continuum hypothesis,

Whitehead’s problem cannot be resolved in ZFC.

30.3 Discussion

J. H. C. Whitehead, motivated by the second Cousin problem, first posed the problem in the 1950s. Stein (1951)answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problemwas considered an important one in algebra for some years.

64

30.4. SEE ALSO 65

Shelah’s result was completely unexpected. While the existence of undecidable statements had been known sinceGödel’s incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hy-pothesis) had all been in pure set theory. The Whitehead problem was the first purely algebraic problem to be provedundecidable.Shelah (1977, 1980) later showed that the Whitehead problem remains undecidable even if one assumes the Contin-uum hypothesis. The Whitehead conjecture is true if all sets are constructible. That this and other statements aboutuncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitiveto the assumed underlying set theory.

30.4 See also• Free abelian group

• Whitehead torsion

• List of statements undecidable in ZFC

• Statements true if all sets are constructible

30.5 References• Eklof, Paul C. (1976), “Whitehead’s Problem is Undecidable”, The American Mathematical Monthly (TheAmericanMathematicalMonthly, Vol. 83, No. 10) 83 (10): 775–788, doi:10.2307/2318684, JSTOR2318684An expository account of Shelah’s proof.

• Eklof, P.C. (2001), “Whitehead problem”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Shelah, S. (1974), “Infinite Abelian groups, Whitehead problem and some constructions”, Israel Journal ofMathematics 18 (3): 243–256, doi:10.1007/BF02757281, MR 0357114

• Shelah, S. (1977), “Whitehead groups may not be free, even assuming CH. I”, Israel Journal of Mathematics28 (3): 193–203, doi:10.1007/BF02759809, MR 0469757

• Shelah, S. (1980), “Whitehead groups may not be free, even assuming CH. II”, Israel Journal of Mathematics35 (4): 257–285, doi:10.1007/BF02760652, MR 0594332

• Stein, Karl (1951), “Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodiz-itätsmoduln und das zweite Cousinsche Problem”,Math. Ann. 123: 201–222, doi:10.1007/BF02054949, MR0043219

Chapter 31

Zero sharp

In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscerniblesand order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (usingGödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC,the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced asa set of formulae in Silver’s 1966 thesis, later published as Silver (1971), where it was denoted by Σ, and rediscoveredby Solovay (1967, p.52), who considered it as a subset of the natural numbers and introduced the notation O# (witha capital letter O; this later changed to a number 0).Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets,while if it does not exist then the universe of all sets is closely approximated by the constructible sets.

31.1 Definition

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constantsymbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentencesabout the constructible universe, with ci interpreted as the uncountable cardinal ℵi. (Here ℵi means ℵi in the fulluniverse, not the constructible universe.)There is a subtlety about this definition: by Tarski’s undefinability theorem it is not in general possible to define thetruth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existenceof a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible todefine the truth of statements about the constructible universe. More generally, the definition of 0# works providedthat there is an uncountable set of indiscernibles for some Lα, and the phrase “0# exists” is used as a shorthand wayof saying this.There are several minor variations of the definition of 0#, which make no significant difference to its properties. Thereare many different choices of Gödel numbering, and 0# depends on this choice. Instead of being considered as a subsetof the natural numbers, it is also possible to encode 0# as a subset of formulae of a language, or as a subset of thehereditarily finite sets, or as a real number.

31.2 Statements that imply the existence of 0#

The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened. The existenceof ω1-Erdős cardinals implies the existence of 0#. This is close to being best possible, because the existence of 0#implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot beused to prove the existence of 0#.Chang’s conjecture implies the existence of 0#.

66

31.3. STATEMENTS EQUIVALENT TO EXISTENCE OF 0# 67

31.3 Statements equivalent to existence of 0#

Kunen showed that 0# exists if and only if there exists a non-trivial elementary embedding for the Gödel constructibleuniverse L into itself.Donald A.Martin and Leo Harrington have shown that the existence of 0# is equivalent to the determinacy of lightfaceanalytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0#.It follows from Jensen’s covering theorem that the existence of 0# is equivalent to ωω being a regular cardinal in theconstructible universe L.Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent tothe existence of 0#.

31.4 Consequences of existence and non-existence

Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L andsatisfies all large cardinal axioms that are realized in L (such as being totally ineffable). It follows that the existenceof 0# contradicts the axiom of constructibility: V = L.If 0# exists, then it is an example of a non-constructible Δ13 set of integers. This is in some sense the simplest possibility for a non-constructible set, since all Σ12 and Π12 sets of integers are constructible.On the other hand, if 0# does not exist, then the constructible universe L is the core model—that is, the canonicalinner model that approximates the large cardinal structure of the universe considered. In that case, Jensen’s coveringlemma holds:

For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the samecardinality as x.

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannotbe removed. For example, consider Namba forcing, that preserves ω1 and collapses ω2 to an ordinal of cofinality ω. Let G be an ω -sequence cofinal on ωL

2 and generic over L. Then no set in L of L-size smaller than ωL2 (which is

uncountable in V, since ω1 is preserved) can cover G , since ω2 is a regular cardinal.

31.5 Other sharps

If x is any set, then x# is defined analogously to 0# except that one uses L[x] instead of L. See the section on relativeconstructibility in constructible universe.

31.6 See also

• 0†, a set similar to 0# where the constructible universe is replaced by a larger inner model with a measurablecardinal.

31.7 References

• 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.

• Harrington, Leo (1978), “Analytic determinacy and 0#", The Journal of Symbolic Logic 43 (4): 685–693,doi:10.2307/2273508, ISSN 0022-4812, MR 518675

68 CHAPTER 31. ZERO SHARP

• 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.

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

• Martin, Donald A. (1970), “Measurable cardinals and analytic games”, Polska Akademia Nauk. FundamentaMathematicae 66: 287–291, ISSN 0016-2736, MR 0258637

• Silver, Jack H. (1971) [1966], “Some applications of model theory in set theory”, Annals of Pure and AppliedLogic 3 (1): 45–110, doi:10.1016/0003-4843(71)90010-6, ISSN 0168-0072, MR 0409188

• Solovay, Robert M. (1967), “A nonconstructible Δ13 set of integers”, Transactions of the American Mathematical Society 127: 50–75, ISSN 0002-9947, JSTOR1994631, MR 0211873

31.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 69

31.8 Text and image sources, contributors, and licenses

31.8.1 Text• Axiom of constructibility Source: https://en.wikipedia.org/wiki/Axiom_of_constructibility?oldid=646125895 Contributors: Michael

Hardy, Schneelocke, Charles Matthews, Maximus Rex, Aleph4, Tobias Bergemann, Lethe, Mboverload, Barnaby dawson, AshtonBenson,Oleg Alexandrov, Penumbra2000, Masnevets, Hmonroe, Hairy Dude, Trovatore, Bota47, SmackBot, Bluebot, Cybercobra, Loadmaster,JRSpriggs, CBM, Jokes Free4Me, Spellmaster, Epsilon0, TrulyBlue, Alexbot, HOOTmag, Addbot, Lightbot, Luckas-bot, Citation bot,VladimirReshetnikov, RedBot, Chricho, ZéroBot, The Nut, ClueBot NG, Loriendrew, Blamelooseradiolabel and Anonymous: 16

• Chang’s model Source: https://en.wikipedia.org/wiki/Chang’{}s_model?oldid=621453134 Contributors: R.e.b., David Eppstein andDeltahedron

• Code (set theory) Source: https://en.wikipedia.org/wiki/Code_(set_theory)?oldid=376574787 Contributors: Charles Matthews, Giftlite,Mets501, JRSpriggs and Hans Adler

• Constructible universe Source: https://en.wikipedia.org/wiki/Constructible_universe?oldid=660903046 Contributors: Zundark, TobyBartels, Edward, Mark Foskey, Schneelocke, Charles Matthews, Dysprosia, Onebyone, Aleph4, Aetheling, Tobias Bergemann, Giftlite,Lethe, Gro-Tsen, Jason Quinn, Ben Standeven, Crisófilax, EmilJ, AshtonBenson, Aleph0~enwiki, Oleg Alexandrov, OwenX, Ryan Reich,BD2412, Rjwilmsi, Tim!, R.e.b., UkPaolo, Aatu, Trovatore, Tsca.bot, NYKevin, Rizome~enwiki, Mets501, Zero sharp, JRSpriggs, CBM,Jokes Free4Me, Gregbard, ProfStevie, Headbomb, JAnDbot, Ling.Nut, Ttwo, Terrek, Sigmundur, DH85868993, Alan U. Kennington,Hotfeba, Neithan Agarwaen, Addbot, Andrewtp21, Lightbot, Yobot, Citation bot, Citation bot 1, Obankston, WikitanvirBot, TannerSwett, Helpful Pixie Bot, PhnomPencil, IkamusumeFan, YFdyh-bot, Kephir, Leonard Huang, Mark viking and Anonymous: 19

• Continuumhypothesis Source: https://en.wikipedia.org/wiki/Continuum_hypothesis?oldid=684645602Contributors: AxelBoldt, ArchibaldFitzchesterfield, Bryan Derksen, Zundark, The Anome, Andre Engels, Hari, Miguel~enwiki, Roadrunner, Shii, Stevertigo, Spiff~enwiki,Michael Hardy, Nixdorf, MartinHarper, Gabbe, Wapcaplet, Karada, Eric119, Snoyes, Marco Krohn, Tim Retout, Schneelocke, Ideyal,Charles Matthews, Timwi, Vanu, Reddi, Paul Stansifer, Doradus, .mau., Phil Boswell, Aleph4, Donarreiskoffer, Fredrik, Bethenco, Tim-rollpickering, Tobias Bergemann, Giftlite, Dbenbenn, Graeme Bartlett, Ævar Arnfjörð Bjarmason, Lethe, Fropuff, Dratman, Eequor,Fuzzy Logic, Icairns, Frenchwhale, Rich Farmbrough, TedPavlic, Guanabot, Luqui, Paul August, EmilJ, Robotje, Reinyday, ObradovicGoran, Haham hanuka, Crust, Jumbuck, Keenan Pepper, Sligocki, Adrian.benko, Oleg Alexandrov, Joriki, StradivariusTV, Ruud Koot,Isnow, Marudubshinki, Graham87, Jobnikon, Yurik, Porcher, Rjwilmsi, NatusRoma, Salix alba, R.e.b., Penumbra2000, FlaBot, JohnBaez, Laubrau~enwiki, YurikBot, Open4D, Ksnortum, Gaius Cornelius, Abarry, Stassats, B-Con, CarlHewitt, SEWilcoBot, Trova-tore, Eltwarg, Doncram, Arthur Rubin, AssistantX, GrinBot~enwiki, SmackBot, Nihonjoe, InverseHypercube, SaxTeacher, TimBent-ley, DHN-bot~enwiki, Tekhnofiend, Cybercobra, DRLB, Byelf2007, Loadmaster, Mets501, MedeaMelana, Quaeler, Dan Gluck, Zerosharp, JRSpriggs, CBM, Discordant~enwiki, Gregbard, Cydebot, Peterdjones, Michael C Price, Thijs!bot, King Bee, Drpixie, Dugwiki,Eleuther, AntiVandalBot, M cuffa, Ling.Nut, Sullivan.t.j, David Eppstein, Kope, R'n'B, Alexcalamaro, Leocat, Ttwo, SpeedOfDark-ness, VolkovBot, Red Act, Nxavar, Layman1, YohanN7, Dogah, SieBot, Ivan Štambuk, Alexis Humphreys, Likebox, Jojalozzo, AnchorLink Bot, CBM2, Cngoulimis, JustinBlank, JuPitEer, Hans Adler, Candhrim~enwiki, Jsondow, Thehelpfulone, MelonBot, Avoided, DOIbot, Unzerlegbarkeit, Lightbot, Know-edu, Legobot, Luckas-bot, Yobot, AnomieBOT, Materialscientist, Citation bot, ArthurBot, Xqbot,RJGray, VladimirReshetnikov, CES1596, Sémaphore, Citation bot 1, Tkuvho, Elockid, RedBot, Belovedeagle, 777sms, Pierpao, Clue-Bot NG, Deadwooddrz, Shivsagardharam, BG19bot, WhatisFGH, Trichometetrahydron, ChrisGualtieri, Ardegloo, Dexbot, Andyhowlett,Qualois, Adam.conkey, Monkbot, Wchargin, MathPhilFan, SoSivr and Anonymous: 115

• Core model Source: https://en.wikipedia.org/wiki/Core_model?oldid=645485084 Contributors: Dmytro, Ben Standeven, Chalst, Salixalba, Trovatore, FlashSheridan, Bluebot, Coremodel, JRSpriggs, Ntsimp, Magioladitis, AnomieBOT, BrideOfKripkenstein, DrilBot, Cn-williams, Mark viking and Anonymous: 3

• Covering lemma Source: https://en.wikipedia.org/wiki/Covering_lemma?oldid=617810862 Contributors: Zundark, Samw, Dysprosia,Dmytro, Oleg Alexandrov, Julien Tuerlinckx, R.e.b., Trovatore, FF2010, SmackBot, Silly rabbit, JRSpriggs, David Eppstein, Kope,Erik9bot, RjwilmsiBot and Anonymous: 1

• Easton’s theorem Source: https://en.wikipedia.org/wiki/Easton’{}s_theorem?oldid=625743396 Contributors: Zundark, Michael Hardy,Charles Matthews, Giftlite, R.e.b., Kompik, RDBury, Mets501, JRSpriggs, CRGreathouse, CBM, Cydebot, Michael C Price, Headbomb,Kope, Addbot, Citation bot, Omnipaedista, Citation bot 1, ZéroBot and Anonymous: 2

• 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

• Goodstein’s theorem Source: https://en.wikipedia.org/wiki/Goodstein’{}s_theorem?oldid=679566169Contributors: AxelBoldt, MichaelHardy, Dominus, Ixfd64, Tim Retout, Charles Matthews, Timwi, Naddy, Merovingian, Henrygb, Tobias Bergemann, Giftlite, Gene WardSmith, Lupin, Gdr, Pmanderson, Gcanyon, Rich Farmbrough, Bender235, Tromp, Aleph0~enwiki, Sligocki, Oleg Alexandrov, OwenX,Orz, Rjwilmsi, R.e.b., Lotu, FlaBot, YurikBot, Trovatore, R.e.s., Arthur Rubin, SmackBot, RDBury, BeteNoir, Eskimbot, TimBentley,Akriasas, Wvbailey, Mets501, 08-15, CBM, Robertinventor, Shlomi Hillel, JAnDbot, Kope, Eliko, TELLME that, Александр Сигачёв,Duncan.Hull, Jsondow, DumZiBoT, Addbot, Luckas-bot, Yobot, Amirobot, Kilom691, LilHelpa, JovanCormac, Tkuvho, SternJacob,Darktower 12345, Durplub, Proof Theorist, Octaazacubane, MaximalIdeal, CitationCleanerBot, Dexbot, JeffAEdmonds, Ynaamad, Cos-mia Nebula, Ianbiringer, Gasole and Anonymous: 38

• Gödel operation Source: https://en.wikipedia.org/wiki/G%C3%B6del_operation?oldid=635363307Contributors: Michael Hardy, R.e.b.,TutterMouse and K9re11

• Innermodel Source: https://en.wikipedia.org/wiki/Inner_model?oldid=607825826Contributors: Zundark,Michael Hardy, CharlesMatthews,Giftlite, Aleph0~enwiki, R.e.b., Mathbot, Trovatore, SmackBot, Viebel, Zero sharp, JRSpriggs, Myasuda, Hans Adler, Addbot, Laaknor-Bot, Yobot, Erik9bot, VS6507, Helpful Pixie Bot, Mark viking and Anonymous: 8

• Inner model theory Source: https://en.wikipedia.org/wiki/Inner_model_theory?oldid=613805448 Contributors: TakuyaMurata, TobiasBergemann, R.e.b., Trovatore, CBM, Gregbard, Qwfp, Locobot, Mark viking and Anonymous: 1

• Jech–Kunen tree Source: https://en.wikipedia.org/wiki/Jech%E2%80%93Kunen_tree?oldid=649445109 Contributors: Michael Hardy,Charles Matthews, Rjwilmsi, R.e.b., Magioladitis, David Eppstein, Wilhelmina Will, PaintedCarpet and Anonymous: 1

70 CHAPTER 31. ZERO SHARP

• Jensen hierarchy Source: https://en.wikipedia.org/wiki/Jensen_hierarchy?oldid=684458070 Contributors: The Anome, Michael Hardy,Delirium, Giftlite, Ben Standeven, R.e.b., SmackBot, Zero sharp, Kope, Robin S, Chimpionspeak, Helpful Pixie Bot, ChrisGualtieri andAnonymous: 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

• 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

• L(R) Source: https://en.wikipedia.org/wiki/L(R)?oldid=635375890 Contributors: Zundark, Schneelocke, Giftlite, Waltpohl, Mysidia,Ben Standeven, Trovatore, Pgk, Bluebot, Ligulembot, Beetstra, JRSpriggs, DOI bot, Citation bot 1, Brirush and Anonymous: 2

• List of statements undecidable inZFC Source: https://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC?oldid=683158211Contributors: AxelBoldt, Zundark, Michael Hardy, Schneelocke, Charles Matthews, Aleph4, Fredrik, Tobias Bergemann, Giftlite, DR-MacIver, Bender235, ZeroOne, Chalst, Aleph0~enwiki, Velella, Uffish, Rjwilmsi, R.e.b., Mathbot, Rbonvall, Trovatore, Jon Awbrey,Syrcatbot, Iridescent, CRGreathouse, CBM, Jokes Free4Me, Pedro Fonini, Headbomb, Mmortal03, David Eppstein, One Night In Hack-ney, Jesin, Yobot, Pcap, Kilom691, AnomieBOT, Cpryby, VladimirReshetnikov, Citation bot 1, Ebony Jackson, ClueBot NG, Pugiator,LinuxIsBetter, K9re11, Monkbot, SoSivr and Anonymous: 23

• Minimalmodel (set theory) Source: https://en.wikipedia.org/wiki/Minimal_model_(set_theory)?oldid=661311108Contributors: Rjwilmsi,R.e.b., JRSpriggs, David Eppstein, Yobot, Citation bot, Citation bot 1 and Trappist the monk

• Mouse (set theory) Source: https://en.wikipedia.org/wiki/Mouse_(set_theory)?oldid=680138484Contributors: Tobias Bergemann, R.e.b.,Trovatore, Coremodel, CBM, Hans Adler, Erik9bot and K9re11

• Naimark’s problem Source: https://en.wikipedia.org/wiki/Naimark’{}s_problem?oldid=598099714 Contributors: GTBacchus, Silver-fish, Charles Matthews, Humus sapiens, ArnoldReinhold, R.e.b., CRGreathouse, Mct mht, Headbomb, Lightbot, Yobot, Ultrawaffle,Riambrid, Leonard Huang and Anonymous: 5

• Paris–Harrington theorem Source: https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem?oldid=666090274 Contrib-utors: Michael Hardy, Bcrowell, Charles Matthews, Tobias Bergemann, Giftlite, Herbee, Gene Nygaard, GregorB, Tizio, Mike Segal,R.e.b., KarlFrei, Trovatore, Arthur Rubin, RDBury, Spireguy, JoshuaZ, Mets501, CBM, Jokes Free4Me, LachlanA, Heysan, Yobot,Chief sequoya, Kozation, Helpful Pixie Bot, Mark viking, תמשל and Anonymous: 12

• Silvermachine Source: https://en.wikipedia.org/wiki/Silver_machine?oldid=625105261Contributors: Michael Hardy, CharlesMatthews,Mboverload, Barnaby dawson, RJHall, Oleg Alexandrov, Rjwilmsi, Friedfish, SmackBot, Jupix, Mets501, Kope, R'n'B, XxTimberlakexx,Kodiologist and Anonymous: 1

• Skolem’s paradox Source: https://en.wikipedia.org/wiki/Skolem’{}s_paradox?oldid=672347617 Contributors: Michael Hardy, Chinju,Graue, TakuyaMurata, Renamed user 4, Charles Matthews, Ruakh, Tobias Bergemann, Tosha, Giftlite, Gdm, Leibniz, Pjacobi, Phiwum,Porton, Nortexoid, Mdd, 4v4l0n42, Noosphere, Spambit, Joriki, Rjwilmsi, R.e.b., Magidin, Chris Pressey, YurikBot, Joth, Trovatore,Mrwright, Curpsbot-unicodify, SmackBot, Imz, Mhss, Joshtrimble, Wvbailey, Loadmaster, Vaughan Pratt, CBM, Myasuda, Aquishix,Maproom, LordAnubisBOT, VanishedUserABC, Dmcq, CBM2, PixelBot, Oshanker, Hans Adler, Hugo Herbelin, Addbot, Unzerleg-barkeit, Lightbot, Yobot, Ptbotgourou, Maltelauridsbrigge, AnomieBOT, Citation bot, Xzungg, Sophus Bie, Citation bot 1, Tkuvho,Trappist the monk, EmausBot, WikitanvirBot, ZéroBot, Tijfo098, Helpful Pixie Bot, Deltahedron, Jochen Burghardt, Hanoch Ben-Yamiand Anonymous: 27

• Statements true in L Source: https://en.wikipedia.org/wiki/Statements_true_in_L?oldid=660956623 Contributors: Chinju, Schnee-locke, Charles Matthews, Tobias Bergemann, Gene Ward Smith, Barnaby dawson, Kundor, JRSpriggs, Jokes Free4Me, R'n'B, Volons,Dexbot and Anonymous: 1

• Strong measure zero set Source: https://en.wikipedia.org/wiki/Strong_measure_zero_set?oldid=644726637 Contributors: AxelBoldtand Michael Hardy

• Suslin’s problem Source: https://en.wikipedia.org/wiki/Suslin’{}s_problem?oldid=679610443Contributors: AxelBoldt, M~enwiki, MichaelHardy, Ixfd64, Chinju, Charles Matthews, Dysprosia, Hyacinth, Dmytro, Tobias Bergemann, Giftlite, Waltpohl, DRMacIver, Barn-aby dawson, Petrus~enwiki, Gauge, Oleg Alexandrov, MFH, Rjwilmsi, Adjusting, R.e.b., Trovatore, Crasshopper, Stotr~enwiki, CR-Greathouse, CBM, Jokes Free4Me, Ntsimp, Thijs!bot, Headbomb, Glivi, David Eppstein, Ttwo, Universityuser, Addbot, DOI bot, Yobot,Gonzalcg, VladimirReshetnikov, FrescoBot, Safinaskar, Citation bot 1, EmausBot, ZéroBot, Bomazi, TheMachine999, Mark viking, Cud-dlyface, Monkbot and Anonymous: 11

• Wetzel’s problem Source: https://en.wikipedia.org/wiki/Wetzel’{}s_problem?oldid=662022578 Contributors: Jason Quinn and DavidEppstein

• Whitehead problem Source: https://en.wikipedia.org/wiki/Whitehead_problem?oldid=680699236 Contributors: AxelBoldt, Zundark,Michael Hardy, BuzzB, Charles Matthews, Dysprosia, Giftlite, DRMacIver, Rhobite, Aleph0~enwiki, Oleg Alexandrov, Rjwilmsi, R.e.b.,Hmonroe, Grubber, Tsiaojian lee, That Guy, From That Show!, Bluebot, JRSpriggs, Gregbard, Ntsimp, Thijs!bot, Addbot, PV=nRT,Citation bot, Locobot, Citation bot 1, Klbrain, Bomazi and Anonymous: 15

• Zero sharp Source: https://en.wikipedia.org/wiki/Zero_sharp?oldid=674929853 Contributors: Michael Hardy, TakuyaMurata, Schnee-locke, Charles Matthews, Dysprosia, Dmytro, Onebyone, Phil Boswell, Aleph4, Peak, Tobias Bergemann, Giftlite, Zeimusu, KlemenKocjancic, Barnaby dawson, Dmeranda, Ben Standeven, Gauge, EmilJ, Oleg Alexandrov, Marudubshinki, Mandarax, R.e.b., Trovatore,Leeannedy, AndrewWTaylor, SmackBot, Chris the speller, Aecea 1, Lambiam, Loadmaster, Zero sharp, JRSpriggs, Vanisaac, CBM,Headbomb, Kope, LokiClock, Yobot, Citation bot, Access Denied, BattyBot, Deltahedron, Blackbombchu and Anonymous: 9

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• File:Undecidable_ZFC_statements_implication_chains.png Source: https://upload.wikimedia.org/wikipedia/en/b/bb/Undecidable_ZFC_statements_implication_chains.png License: CC0 Contributors:Own workOriginal artist:LinuxIsBetter

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