Elements of Set Theory - CM0832 - EAFIT · Elements of Set Theory - CM0832 ... The objects are called elements (members) of the set.” [Hrbacek and Jech 1999, ... [Enderton 1977]

Elements of Set Theory - CM0832(complementary slides)

Andrés Sicard-Ramírez

Universidad EAFIT

Semester 2017-2

Administrative InformationCourse web pagehttp://www1.eafit.edu.co/asr/courses/set-theory-CM0832/

Exams, bibliography, etc.See course web page.

Introduction 2/144

http://www1.eafit.edu.co/asr/courses/set-theory-CM0832/

NotationLogical constants∧ (and) conjunction∨ (or) inclusive∗ disjunction⇒ (if , then ) implication¬ (not) negation⇔ (if and only if, iff) bi-implication, equivalence⊥ (falsity) bottom, falsum∀𝑥 (for every 𝑥) universal quantifier∃𝑥 (there exists a 𝑥) existential quantifier

SetsSets will be denote by lowercase letters (𝑎, 𝑏, …), uppercase letters(𝐴, 𝐵, …), script letters (𝒜, ℬ, …) and Greek letters (𝛼, 𝛽, …).

∗One or the other or both.Introduction 3/144

Origins

Georg Cantor (1845 – 1918)

Introduction 4/144

Origins

“Set theory was invented by Georg Can-tor…It was however Cantor who realizedthe significance of one-to-one functionsbetween sets and introduced the notionof cardinality of a set.” (p. 15)

Introduction 5/144

Origins

“Set theory was born on that Decem-ber 1873 day when Cantor establishedthat the reals are uncountable, i.e.there is no one-to-one correspond-ence between the reals and the naturalnumbers.” (p. XII)

Introduction 6/144

Naive Set TheoryRemarkCantor set theory is also called naive set theory.

Cantor’s set definitionCantor (1895, p. 1):

“Unter einer Menge verstehen wir jede Zusammenfassung M vonbestimmten wohlunterschiedenen Objekten m unserer Anschauungoder unseres Denkens (welche die Elemente von M genannt wer-den) zu einem Ganzen.”

Translation (Hrbacek and Jech 1999, p. 1):“A set is a collection into a whole of definite, distinct objectsof our intuition or our thought. The objects are called elements(members) of the set.”

Introduction 7/144

Naive Set TheoryRemarkCantor set theory is also called naive set theory.

Cantor’s set definitionCantor (1895, p. 1):

“Unter einer Menge verstehen wir jede Zusammenfassung M vonbestimmten wohlunterschiedenen Objekten m unserer Anschauungoder unseres Denkens (welche die Elemente von M genannt wer-den) zu einem Ganzen.”

Translation (Hrbacek and Jech 1999, p. 1):“A set is a collection into a whole of definite, distinct objectsof our intuition or our thought. The objects are called elements(members) of the set.”

Introduction 8/144

Naive Set Theory

Membership relation (a binary relation)𝑡 ∈ 𝐴 means that 𝑡 is a member of 𝐴.The expression 𝑡 ∉ 𝐴 is an abbreviation of ¬(𝑡 ∈ 𝐴).

Principle of extensionality, empty set, pair set, union and intersection,subset and power setSee whiteboard.

Introduction 9/144

Naive Set Theory

Membership relation (a binary relation)𝑡 ∈ 𝐴 means that 𝑡 is a member of 𝐴.The expression 𝑡 ∉ 𝐴 is an abbreviation of ¬(𝑡 ∈ 𝐴).

Principle of extensionality, empty set, pair set, union and intersection,subset and power setSee whiteboard.

Introduction 10/144

Naive Set TheoryProblemImplicit use of properties of sets.

Example (axiom of choice)

Illustration of the axiom.∗

The proof that a real function 𝑓 is continuous at a point 𝑝 if and only if 𝑓is sequentially continuous at 𝑝 [Heine 1872] requires of the axiom ofchoice. See (Moore 1982, p. 14) or (Hrbacek and Jech 1999, pp. 145-6).

∗Figure from https://commons.wikimedia.org/w/index.php?curid=48219447.Introduction 11/144

https://commons.wikimedia.org/w/index.php?curid=48219447













Naive Set TheoryProblemToo general method of abstraction.

Example (Russell’s paradox)See whiteboard.

Introduction 14/144


Example (Russell’s paradox)See whiteboard.

Introduction 15/144

Russell’s Paradox

Gottlob Frege(1848 – 1925)

Bertrand Russell(1872 – 1970)

Introduction 16/144

Letter from Russell to van Heijenoort∗

Penrhyndeudraeth, 23 November 1962Dear Professor van Heijenoort,As I think about acts of integrity and grace, I realise there is nothing in myknowledge to compare with Frege’s dedication to truth. His entire life’swork was on the verge of completion, much of his work had been ignoredto the benefit of men infinitely less capable, his second volume was aboutto be published, and upon finding that his fundamental assumption was inerror, he responded with intellectual pleasure clearly submerging anyfeelings of personal disappointment. It was almost superhuman and atelling indication of that of which men are capable if their dedication is tocreative work and knowledge instead of cruder efforts to dominate and beknown.

Yours sincerelyBertrand Russell

∗van Heijenoort (1967, p. 127).Introduction 17/144


ExerciseWhich is the paradox called Berry paradox?

Introduction 18/144

Informally Building Sets∗

Sets-An Informal View 7

2. Show that no two o~e three sets 0, {0}, and {{0}} are equal to each other.

3. Show that if B S; C, then f!J B S; f!Jc.

4. Assume that x and yare members ofa set B. Show that {{x}, {x, y}} E f!Jf!JB.

SETS-AN INFORMAL VIEW

We are about to present a somewhat vague description of how sets are obtained. (The description will be repeated much later in precise form.) None of our later work will actually depend on this informal description, but we hope it will illuminate the motivation behind some of the things we will do.

IX

w

o

Fig.2. Vo is the set A of atoms.

First we gather together all those things that are not themselves sets but that we want to have as members of sets. Call such things atoms. For example, if we want to be able to speak of the set of all two-headed coins, then we must include all such coins in our collection of atoms. Let A be the set of all atoms; it is the first set in our description.

We now proceed to build up a hierarchy

Vos; VlS; V2

s;",

of sets. At the bottom level (in a vertical arrangement as in Fig. 2) we take Vo = A, the set of atoms. the next level will also contain all sets of atoms:

Vl = Vo u f!J Vo = A u f!J A.

𝑉0 = 𝐴 (set of atoms) 𝑉𝜔 = 𝑉0 ∪ 𝑉1 ∪ ⋯ 𝑉𝛼+1 = 𝑉𝛼 ∪ 𝒫𝑉𝛼𝑉𝑛+1 = 𝑉𝑛 ∪ 𝒫𝑉𝑛 𝑉𝜔+1 = 𝑉𝜔 ∪ 𝒫𝑉𝜔∗Fig. 2 (Enderton 1977).

Introduction 19/144

Informally Building Sets∗

Sets-An Informal View 9

is that the atoms serve no mathematically necessary purpose, so we banish them; we take A = 0. In so doing, we lose the ability to form sets of flowers or sets of people. But this is no cause for concern; we do not need set theory to talk about people and we do not need people in our set theory. But we definitely do want to have sets of numbers, e.g., {2, 3 + in}. Numbers do not appear at first glance to be sets. But as we shall discover (in Chapters 4 and 5), we can find sets that serve perfectly well as numbers.

Our theory then will ignore all objects that are not sets (as interesting and real as such objects may be). Instead we will concentrate just on "pure" sets that can be constructed without the use of such external objects. In

1 v. 1

v.,

I Fig. 3. The ordinals are the backbone of the universe.

particular, any member of one of our sets will itself be a set, and each of its members, if any, will be a set, and so forth. (This does not produce an infinite regress, because we stop when we reach 0.)

Now that we have banished atoms, the picture becomes narrower (Fig. 3). The construction is also simplified. We have defined ~+1 to be J<. u £!l>V . Now it turns out that this is the same as A u £!l>V (see Exercise

a a 6). With A = 0, we have simply ~+1 = £!l>~.

Exercises

5. Define the rank of a set c to be the least IX such that c £ V . Compute the rank of {{0}}. Compute the rank of {0, {0}, {0, {0}}}. a

6. We have stated that ~ + 1 = A u £!l> ~. Prove this at least for IX < 3.

7. List all the members of V3

• List all the members of V4

• (It is to be assumed here that there are no atoms.)

The ordinal numbers are the backbone of the universe.

𝑉0 = ∅ (no atoms) 𝑉𝛼+1 = 𝒫𝑉𝛼∗Fig. 3 (Enderton 1977).

Introduction 20/144

ClassesInformal descriptionA set is a class, but some classes are too large to be a sets.

ExampleThe collection of all sets.

RemarkA class A is a set if 𝐴 ⊆ 𝑉𝛼 (𝐴 ∈ 𝑉𝛼+1) for some 𝛼.

Introduction 21/144

Axiomatic MethodSome features

Axioms: Explicitly list of assumptions

Theorems: Logical consequence of the axiomsProperty of set theory: It should be an axiom or a theorem

Introduction 22/144


Axioms: Explicitly list of assumptionsTheorems: Logical consequence of the axioms

Property of set theory: It should be an axiom or a theorem

Introduction 23/144


Axioms: Explicitly list of assumptionsTheorems: Logical consequence of the axiomsProperty of set theory: It should be an axiom or a theorem

Introduction 24/144

Axiomatic MethodAxiomatic set theory as a fundational system for mathematics

A common affirmation is that “mathematics can be embedded into settheory”.

“our axioms provide a sufficient collection of assumptions for the devel-opment of the whole of mathematics—a remarkable fact.” (Enderton1977, p. 11)

“Experience has shown that practically all notions used in contempor-ary mathematics can be defined, and their mathematical propertiesderived, in this axiomatic system. In this sense, the axiomatic settheory serves as a satisfactory foundations for the other branches ofmathematics.” (Hrbacek and Jech 1999, p. 3)

Introduction 25/144

Axiomatic MethodSome axiomatic systems

Zermelo-Fraenkel set theory with choice (ZFC)von Neumann–Bernays–Gödel set theoryTarski–Grothendieck set theory

Introduction 26/144

Axiomatic MethodFirst-Order Theories∗

“The adjective ‘first-order’ is used to distinguish the languages…from thosein which are predicates having other predicates or functions as arguments,or quantification over functions or predicates, or both.” (Mendelson 1997,p. 56)

Primitive notionsWe only need two primitive notions, “set” and “member”.

∗For an introduction to first-order languages and first-order theories see, for example,(Hamilton 1978) or (Mendelson 1997).

Introduction 27/144

Axiomatic MethodFirst-Order Theories∗

“The adjective ‘first-order’ is used to distinguish the languages…from thosein which are predicates having other predicates or functions as arguments,or quantification over functions or predicates, or both.” (Mendelson 1997,p. 56)

Primitive notionsWe only need two primitive notions, “set” and “member”.

∗For an introduction to first-order languages and first-order theories see, for example,(Hamilton 1978) or (Mendelson 1997).

Introduction 28/144

Axioms and Operations

Extensionality AxiomExtensionality axiomIf two sets have exactly the same members, then they are equal, that is,

∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].

RemarkA set is pure if its members are also sets.

QuestionHave we any set? No, we haven’t.

Axioms and Operations 30/144


∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].RemarkA set is pure if its members are also sets.




∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].RemarkA set is pure if its members are also sets.



Some Axioms for Building Sets

Empty (existence) axiomThere is a set having no members, that is,

∃𝐵 ∀𝑥 (𝑥 ∉ 𝐵).

Pairing axiomFor any sets 𝑢 and 𝑣, there is a set having as members just 𝑢 and 𝑣, thatis,

∀𝑎 ∀𝑏 ∃𝐶 ∀𝑥 (𝑥 ∈ 𝐶 ⇔ 𝑥 = 𝑎 ∨ 𝑥 = 𝑏).



Empty (existence) axiomThere is a set having no members, that is,

∃𝐵 ∀𝑥 (𝑥 ∉ 𝐵).Pairing axiomFor any sets 𝑢 and 𝑣, there is a set having as members just 𝑢 and 𝑣, thatis,

∀𝑎 ∀𝑏 ∃𝐶 ∀𝑥 (𝑥 ∈ 𝐶 ⇔ 𝑥 = 𝑎 ∨ 𝑥 = 𝑏).



Union axiom (first version)For any sets 𝑎 and 𝑏, there is a set whose members are those sets belongingeither to 𝑎 or to 𝑏 (or both), that is,

∀𝑎 ∀𝑏 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏).Power set axiomFor any set 𝑎, there is a set whose members are exactly the subsets of 𝑎,that is,

∀𝑎 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ⊆ 𝑎),where

𝑢 ⊆ 𝑣 ⇔ ∀𝑡 (𝑡 ∈ 𝑢 ⇒ 𝑡 ∈ 𝑣).


Subset Axiom SchemeIntroductionSee whiteboard.

Abstractions from the empty, pairing, union and power set axiomsLet 𝑎, 𝑏, 𝑢 and 𝑣 be sets, then

∅ = {𝑥 ∣ 𝑥 ≠ 𝑥},{𝑢, 𝑣} = {𝑥 ∣ 𝑥 = 𝑢 ∨ 𝑥 = 𝑣},𝑎 ∪ 𝑏 = {𝑥 ∣ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏},

𝒫𝑎 = {𝑥 ∣ 𝑥 ⊆ 𝑎}.


Subset Axiom SchemeIntroductionSee whiteboard.

Abstractions from the empty, pairing, union and power set axiomsLet 𝑎, 𝑏, 𝑢 and 𝑣 be sets, then

∅ = {𝑥 ∣ 𝑥 ≠ 𝑥},{𝑢, 𝑣} = {𝑥 ∣ 𝑥 = 𝑢 ∨ 𝑥 = 𝑣},𝑎 ∪ 𝑏 = {𝑥 ∣ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏},

𝒫𝑎 = {𝑥 ∣ 𝑥 ⊆ 𝑎}.


Subset Axiom Scheme

Subset axiom scheme (axiom scheme of comprehension/separation)For each formula 𝜑, not containing 𝐵, the following is an axiom:

∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).

RemarkWe stated an axiom scheme!

Abstraction from the subset axiom scheme{𝑥 ∈ 𝑐 ∣ 𝜑} is the set of all 𝑥 ∈ 𝑐 satisfying the property 𝜑.


Subset Axiom Scheme


∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).RemarkWe stated an axiom scheme!



Subset Axiom Scheme


∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).RemarkWe stated an axiom scheme!



Subset Axiom Scheme

Theorem 2A (Enderton 1977)There is no set to which every set belongs.

ExerciseWhy does the subset axiom scheme avoid the Berry paradox?


Subset Axiom Scheme

Theorem 2A (Enderton 1977)There is no set to which every set belongs.

ExerciseWhy does the subset axiom scheme avoid the Berry paradox?


Arbitrary Unions and IntersectionsDefinitionLet 𝐴 be a set. The union ⋃ 𝐴 of 𝐴 is defined by

⋃ 𝐴 = {𝑥 ∣ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏}.Examples

𝑎 ∪ 𝑏 = ⋃{𝑎, 𝑏},⋃{𝑎} = 𝑎,

⋃ ∅ = ∅.

Union axiom (final version)For any set 𝐴, there exists a set 𝐵 whose elements are exactly the membersof the members of 𝐴, that is,

∀𝐴 ∃𝐵 ∀𝑥 [𝑥 ∈ 𝐵 ⇔ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏)].


Arbitrary Unions and IntersectionsDefinitionLet 𝐴 be a set. The union ⋃ 𝐴 of 𝐴 is defined by

⋃ 𝐴 = {𝑥 ∣ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏}.Examples

𝑎 ∪ 𝑏 = ⋃{𝑎, 𝑏},⋃{𝑎} = 𝑎,

⋃ ∅ = ∅.

Union axiom (final version)For any set 𝐴, there exists a set 𝐵 whose elements are exactly the membersof the members of 𝐴, that is,

∀𝐴 ∃𝐵 ∀𝑥 [𝑥 ∈ 𝐵 ⇔ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏)].


Arbitrary Unions and Intersections

Theorem 2B (Enderton 1977)For any nonempty set 𝐴, there exists a unique set 𝐵 such that for any 𝑥,

𝑥 ∈ 𝐵 ⇔ 𝑥 belongs to every member of 𝐴.

CharacterisationLet 𝐴 be a nonempty set. The intersecction ⋂ 𝐴 of 𝐴 can be character-ised by

⋂ 𝐴 = {𝑥 ∣ ∀𝑏 (𝑏 ∈ 𝐴 ⇒ 𝑥 ∈ 𝑏}, for𝐴 ≠ ∅.


Arbitrary Unions and Intersections

Theorem 2B (Enderton 1977)For any nonempty set 𝐴, there exists a unique set 𝐵 such that for any 𝑥,

𝑥 ∈ 𝐵 ⇔ 𝑥 belongs to every member of 𝐴.

CharacterisationLet 𝐴 be a nonempty set. The intersecction ⋂ 𝐴 of 𝐴 can be character-ised by

⋂ 𝐴 = {𝑥 ∣ ∀𝑏 (𝑏 ∈ 𝐴 ⇒ 𝑥 ∈ 𝑏}, for𝐴 ≠ ∅.


Algebra of Sets

Exercise (Enderton (1977), Exercise 2.18)Assume that 𝐴 and 𝐵 are subsets of 𝑆. List all of the different sets thatcan be made from these three by use of the binary operations ∪, ∩, and −.

The Venn diagram shows four possible regions for shading, that is, wehave 24 different sets given by∅, 𝐴, 𝐵, 𝑆, 𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵, 𝐴 − 𝐵, 𝐵 − 𝐴, 𝐴 + 𝐵, 𝑆 − 𝐴, 𝑆 − 𝐵,𝑆 − (𝐴 ∪ 𝐵), 𝑆 − (𝐴 ∩ 𝐵), 𝑆 − (𝐴 − 𝐵), 𝑆 − (𝐵 − 𝐴) and 𝑆 − (𝐴 + 𝐵),where the binary operation + is the symmetric difference defined by

𝐴 + 𝐵 = (𝐴 − 𝐵) ∪ (𝐵 − 𝐴)= (𝐴 ∪ 𝐵) − (𝐴 ∩ 𝐵).


Algebra of Sets

Exercise (Enderton (1977), Exercise 2.18)Assume that 𝐴 and 𝐵 are subsets of 𝑆. List all of the different sets thatcan be made from these three by use of the binary operations ∪, ∩, and −.The Venn diagram shows four possible regions for shading, that is, wehave 24 different sets given by∅, 𝐴, 𝐵, 𝑆, 𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵, 𝐴 − 𝐵, 𝐵 − 𝐴, 𝐴 + 𝐵, 𝑆 − 𝐴, 𝑆 − 𝐵,𝑆 − (𝐴 ∪ 𝐵), 𝑆 − (𝐴 ∩ 𝐵), 𝑆 − (𝐴 − 𝐵), 𝑆 − (𝐵 − 𝐴) and 𝑆 − (𝐴 + 𝐵),where the binary operation + is the symmetric difference defined by

𝐴 + 𝐵 = (𝐴 − 𝐵) ∪ (𝐵 − 𝐴)= (𝐴 ∪ 𝐵) − (𝐴 ∩ 𝐵).


Relations and Functions

Ordered PairsDefinitionLet 𝑎 and 𝑏 be sets. The order pair ⟨𝑎, 𝑏⟩ should be a set such that

⟨𝑎, 𝑏⟩ = ⟨𝑐, 𝑑⟩ ⇔ 𝑎 = 𝑐 ∧ 𝑏 = 𝑑.

We can define the order pairs via Kuratowski’s definition, that is, ⟨𝑎, 𝑏⟩ isthe set {{𝑎}, {𝑎, 𝑏}}.

ExerciseTo give a different definition of order pairs.

Relations and Functions 50/144

Cartesian ProductDefinitionLet 𝐴 and 𝐵 be sets. The Cartesian product of 𝐴 and 𝐵, denoted by 𝐴×𝐵,is defined by

𝐴 × 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵}.

Is 𝐴 × 𝐵 a set?Yes! We can define it via the subset axiom scheme by

𝐴 × 𝐵 = {⟨𝑥, 𝑦⟩ ∈ 𝒫𝒫(𝐴 ∪ 𝐵) ∣ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵}.


RelationsDefinitionA relation is a set of ordered pairs.

ExamplesSee whiteboard.


RelationsDefinitionLet 𝑅 be a relation. We define the domain, the range and the field of 𝑅 by

dom 𝑅 = {𝑥 ∣ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝑅)},ran 𝑅 = {𝑦 ∣ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ 𝑅)},fld 𝑅 = dom 𝑅 ∪ ran 𝑅.

Are dom 𝑅 and ran 𝑅 sets?Yes! We can define them via the subset axiom scheme by

dom 𝑅 = {𝑥 ∈ ⋃ ⋃ 𝑅 ∣ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝑅)},ran 𝑅 = {𝑦 ∈ ⋃ ⋃ 𝑅 ∣ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ 𝑅)}.


RelationsDefinitionLet 𝑅 be a relation. We define the domain, the range and the field of 𝑅 by

dom 𝑅 = {𝑥 ∣ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝑅)},ran 𝑅 = {𝑦 ∣ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ 𝑅)},fld 𝑅 = dom 𝑅 ∪ ran 𝑅.

Are dom 𝑅 and ran 𝑅 sets?Yes! We can define them via the subset axiom scheme by

dom 𝑅 = {𝑥 ∈ ⋃ ⋃ 𝑅 ∣ ∃𝑦(⟨𝑥, 𝑦⟩ ∈ 𝑅)},ran 𝑅 = {𝑦 ∈ ⋃ ⋃ 𝑅 ∣ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ 𝑅)}.


FunctionsDefinitionA function is a relation 𝐹 such that for each 𝑥 in dom 𝐹 there is only one 𝑦such that 𝑥𝐹𝑦.

Exercise (Enderton (1977), Exercise 3.11)Prove the following version (for functions) of the extensionality principle:Assume that 𝐹 and 𝐺 are functions, dom 𝐹 = dom 𝐺, and 𝐹(𝑥) = 𝐺(𝑥)for all 𝑥 in the common domain. Then 𝐹 = 𝐺.


FunctionsDefinitionA function is a relation 𝐹 such that for each 𝑥 in dom 𝐹 there is only one 𝑦such that 𝑥𝐹𝑦.

Exercise (Enderton (1977), Exercise 3.11)Prove the following version (for functions) of the extensionality principle:Assume that 𝐹 and 𝐺 are functions, dom 𝐹 = dom 𝐺, and 𝐹(𝑥) = 𝐺(𝑥)for all 𝑥 in the common domain. Then 𝐹 = 𝐺.


FunctionsDefinitionsLet 𝐴, 𝐹 and 𝐺 be sets. We define, the inverse of 𝐹 , the compositionof 𝐹 and 𝐺, the restriction of 𝐹 to 𝐴 and the image of 𝐴 under 𝐹 by

𝐹 −1 = {⟨𝑦, 𝑥⟩ ∣ 𝑥𝐹𝑦} (inverse of 𝐹 )

𝐹 ∘ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑡 (𝑥𝐺𝑡 ∧ 𝑡𝐹𝑦)} (composition of 𝐹 and 𝐺)

𝐹 ↾ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦 ∧ 𝑥 ∈ 𝐴} (restriction of 𝐹 to 𝐴)

𝐹⟦𝐴⟧ = ran(𝐹 ↾ 𝐴) (image of 𝐴 under 𝐹 )= {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦}


Functions

Exercise (Enderton (1977), Exercise 3.18)Let 𝑅 be the set

{⟨0, 1⟩, ⟨0, 2⟩, ⟨0, 3⟩, ⟨1, 2⟩, ⟨1, 3⟩, ⟨2, 3⟩}.

To find 𝑅 ∘ 𝑅, 𝑅 ↾ {1}, 𝑅−1 ↾ {1}, 𝑅⟦{1}⟧ and 𝑅−1⟦{1}⟧.

Exercise (Enderton (1977), Exercise 3.18)Let 𝐴, 𝐹 and 𝐺 be sets. Show that 𝐹 −1, 𝐹 ∘ 𝐺, 𝐹 ↾ 𝐴 and 𝐹⟦𝐴⟧ are sets.


Functions

Exercise (Enderton (1977), Exercise 3.18)Let 𝑅 be the set

{⟨0, 1⟩, ⟨0, 2⟩, ⟨0, 3⟩, ⟨1, 2⟩, ⟨1, 3⟩, ⟨2, 3⟩}.

To find 𝑅 ∘ 𝑅, 𝑅 ↾ {1}, 𝑅−1 ↾ {1}, 𝑅⟦{1}⟧ and 𝑅−1⟦{1}⟧.

Exercise (Enderton (1977), Exercise 3.18)Let 𝐴, 𝐹 and 𝐺 be sets. Show that 𝐹 −1, 𝐹 ∘ 𝐺, 𝐹 ↾ 𝐴 and 𝐹⟦𝐴⟧ are sets.


Axiom of Choice

Axiom of choice (first form)For any relation 𝑅 there is a function 𝐻 ⊆ 𝑅 with dom 𝐻 = dom 𝑅.

RemarkIs the axiom of choice accepted in constructive mathematics? (See, forexample, Martin-Löf (2006)).


Axiom of Choice

Axiom of choice (first form)For any relation 𝑅 there is a function 𝐻 ⊆ 𝑅 with dom 𝐻 = dom 𝑅.

RemarkIs the axiom of choice accepted in constructive mathematics? (See, forexample, Martin-Löf (2006)).


FunctionsDefinitionLet 𝐴 and 𝐵 be sets. We define the set of functions 𝐹 from 𝐴 into 𝐵 by

𝐴𝐵 = 𝐵𝐴 = {𝐹 ∣ 𝐹 ∶ 𝐴 → 𝐵}.

Examples{0, 1}𝜔

∅𝐴 = ∅ for 𝐴 ≠ ∅ (any function can have a non-empty domain and anempty range)𝐴∅ = {∅} for any set 𝐴 (∅ is the only function with an empty domain)

Is 𝐵𝐴 a set?Yes! We can define it via the subset axiom scheme by

𝐵𝐴 = {𝐹 ∈ 𝒫(𝐴 × 𝐵) ∣ 𝐹 ∶ 𝐴 → 𝐵}.



𝐴𝐵 = 𝐵𝐴 = {𝐹 ∣ 𝐹 ∶ 𝐴 → 𝐵}.Examples

{0, 1}𝜔



𝐵𝐴 = {𝐹 ∈ 𝒫(𝐴 × 𝐵) ∣ 𝐹 ∶ 𝐴 → 𝐵}.




{0, 1}𝜔

∅𝐴 = ∅ for 𝐴 ≠ ∅ (any function can have a non-empty domain and anempty range)

𝐴∅ = {∅} for any set 𝐴 (∅ is the only function with an empty domain)


𝐵𝐴 = {𝐹 ∈ 𝒫(𝐴 × 𝐵) ∣ 𝐹 ∶ 𝐴 → 𝐵}.




{0, 1}𝜔



𝐵𝐴 = {𝐹 ∈ 𝒫(𝐴 × 𝐵) ∣ 𝐹 ∶ 𝐴 → 𝐵}.




{0, 1}𝜔



𝐵𝐴 = {𝐹 ∈ 𝒫(𝐴 × 𝐵) ∣ 𝐹 ∶ 𝐴 → 𝐵}.


Equivalence RelationsIdea∗

56 3. Relations and Functions

bricks.) This changes the picture (Fig. 12c); each box is now, in our mind, a single point. The set B of boxes is very different from the set A. In our example, B has only six members whereas A is infinite. (When we get around to defining "six" and "infinite" officially, we must certainly do it in a way that makes the preceding sentence true.)

The process of transforming a situation like Fig. 12a into Fig. 12c is common in abstract algebra and elsewhere in mathematics. And in Chapter 5 the process will be applied several times in the construction of the real numbers.

Suppose we now define a binary relation R on A as follows: For x and y in A,

xRy ¢> x and yare in the same little box.

• • •

• • •

(a) (b) (c)

Fig. 12. Partitioning a set into six little boxes.

Then we can easily see that R has the following three properties.

1. R is reflexive on A, by which we mean that xRx for all x E A. 2. R is symmetric, by which we mean that whenever xRy, then also

yRx. 3. R is transitive, by which we mean that whenever xRy and yRz,

then also xRz.

Definition R is an equivalence relation on A iff R is a binary relation on A that is reflexive on A, symmetric, and transitive.

Theorem 3M If R is a symmetric and transitive relation, then R is an equivalence relation on fld R.

Proof Any relation R is a binary relation on its field, since

R ~ dom R x ran R ~ fld R x fld R.

What we must show is that R is reflexive on fld R. We have

X E dom R => xRy

=> xRy & yRx

=> xRx

for some y

by symmetry

by transitivity,

and a similar calculation applies to points in ran R.

DefinitionsLet 𝑅 be a binary relation on a set 𝐴. The relation 𝑅 is

reflexive iff 𝑥𝑅𝑥 for all 𝑥 ∈ 𝐴,symmetric iff 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 andtransitive iff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ⇒ 𝑥𝑅𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴.

∗Fig. 12 (Enderton 1977).Relations and Functions 67/144







• • •

• • •

(a) (b) (c)





then also xRz.






X E dom R => xRy

=> xRy & yRx

=> xRx

for some y

by symmetry

by transitivity,



reflexive iff 𝑥𝑅𝑥 for all 𝑥 ∈ 𝐴,

symmetric iff 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 andtransitive iff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ⇒ 𝑥𝑅𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴.








• • •

• • •

(a) (b) (c)





then also xRz.






X E dom R => xRy

=> xRy & yRx

=> xRx

for some y

by symmetry

by transitivity,



reflexive iff 𝑥𝑅𝑥 for all 𝑥 ∈ 𝐴,symmetric iff 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 and

transitive iff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ⇒ 𝑥𝑅𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴.








• • •

• • •

(a) (b) (c)





then also xRz.






X E dom R => xRy

=> xRy & yRx

=> xRx

for some y

by symmetry

by transitivity,



reflexive iff 𝑥𝑅𝑥 for all 𝑥 ∈ 𝐴,symmetric iff 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 andtransitive iff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ⇒ 𝑥𝑅𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴.


Equivalence RelationsDefinitionLet 𝑅 be a binary relation on a set 𝐴. The relation 𝑅 is a equivalencerelation iff 𝑅 is reflexive, symmetric and transitive.

QuestionsLet 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalencerelation?Let 𝐴 ≠ ∅ be a set. Is the relation ∅ on 𝐴 an equivalence relation?Let 𝐴 be a set. Is the relation 𝐴 × 𝐴 an equivalence relations?Let 𝐴 be a singleton. It is possible to define an equivalence relationon 𝐴?



QuestionsLet 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalencerelation?

Let 𝐴 ≠ ∅ be a set. Is the relation ∅ on 𝐴 an equivalence relation?Let 𝐴 be a set. Is the relation 𝐴 × 𝐴 an equivalence relations?Let 𝐴 be a singleton. It is possible to define an equivalence relationon 𝐴?



QuestionsLet 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalencerelation?Let 𝐴 ≠ ∅ be a set. Is the relation ∅ on 𝐴 an equivalence relation?

Let 𝐴 be a set. Is the relation 𝐴 × 𝐴 an equivalence relations?Let 𝐴 be a singleton. It is possible to define an equivalence relationon 𝐴?



QuestionsLet 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalencerelation?Let 𝐴 ≠ ∅ be a set. Is the relation ∅ on 𝐴 an equivalence relation?Let 𝐴 be a set. Is the relation 𝐴 × 𝐴 an equivalence relations?

Let 𝐴 be a singleton. It is possible to define an equivalence relationon 𝐴?



QuestionsLet 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}. Is the equality relation on 𝐴 an equivalencerelation?Let 𝐴 ≠ ∅ be a set. Is the relation ∅ on 𝐴 an equivalence relation?Let 𝐴 be a set. Is the relation 𝐴 × 𝐴 an equivalence relations?Let 𝐴 be a singleton. It is possible to define an equivalence relationon 𝐴?


Linear Ordering RelationsMotivationWhat means that 𝑅 is an ordering relation on a set 𝐴?

DefinitionLet 𝑅 be a binary relation on a set 𝐴. The relation 𝑅 satisfies trichotomyif exactly one of the three alternatives

𝑥𝑅𝑦, 𝑥 = 𝑦 or 𝑦𝑅𝑥

holds for all for all 𝑥, 𝑦 ∈ 𝐴.


Linear Ordering RelationsMotivationWhat means that 𝑅 is an ordering relation on a set 𝐴?

DefinitionLet 𝑅 be a binary relation on a set 𝐴. The relation 𝑅 satisfies trichotomyif exactly one of the three alternatives

𝑥𝑅𝑦, 𝑥 = 𝑦 or 𝑦𝑅𝑥

holds for all for all 𝑥, 𝑦 ∈ 𝐴.


Linear Ordering RelationsDefinitionLet 𝐴 be a set. A linear ordering (or total ordering) on 𝐴 is a binaryrelation 𝑅 on 𝐴 such that:

1. 𝑅 is transitive relation and2. 𝑅 satisfies trichotomy on 𝐴.

Example

0123

⋮


Linear Ordering RelationsDefinitionLet 𝐴 be a set. A linear ordering (or total ordering) on 𝐴 is a binaryrelation 𝑅 on 𝐴 such that:

1. 𝑅 is transitive relation and2. 𝑅 satisfies trichotomy on 𝐴.

Example

0123

⋮


Natural Numbers

Natural NumbersApproaches for introducing mathematical objects

axiomaticdefinitional

Definitional approach for introducing natural numbersWe shall define natural numbers in terms of setsWe shall prove the properties of natural numbers from properties ofsets

How to define natural numbers in terms of sets?

Natural Numbers 81/144











Natural Numbersvon Neumann’s constructionInformal idea: A natural number is the set of all smaller natural numbers

0 = ∅,1 = {0} = {∅},2 = {0, 1} = {∅, {∅}},3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}},

⋮

Some “extra” properties0 ∈ 1 ∈ 2 ∈ 3 ∈ ...

0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ ...


Natural Numbersvon Neumann’s constructionInformal idea: A natural number is the set of all smaller natural numbers

0 = ∅,1 = {0} = {∅},2 = {0, 1} = {∅, {∅}},3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}},

⋮Some “extra” properties

0 ∈ 1 ∈ 2 ∈ 3 ∈ ...

0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ ...


Natural NumbersA wrong impredicative definition

𝑛 = {0, 1, … , 𝑛 − 1}.“We cannot just say that a set 𝑛 is a natural number if its elements are allthe smaller natural numbers, because such a “definition” would involve thevery concept being defined.” (Hrbacek and Jech 1999, p. 40)


Natural NumbersDefinitionLet 𝑎 be a set. The successor of 𝑎 is

𝑎+ = 𝑎 ∪ {𝑎}.

Examples

0 = ∅,1 = ∅+,2 = ∅++,3 = ∅+++,⋮


Natural NumbersDefinitionLet 𝑎 be a set. The successor of 𝑎 is

𝑎+ = 𝑎 ∪ {𝑎}.

Examples

0 = ∅,1 = ∅+,2 = ∅++,3 = ∅+++,⋮


Natural NumbersDefinitionA set 𝐴 is called inductive iff

∅ ∈ 𝐴 andif 𝑎 ∈ 𝐴 then 𝑎+ ∈ 𝐴.

RemarkAn inductive set will be an infinite set.

QuestionAre there inductive sets?

Infinity axiomThere exists an inductive set, that is,

∃𝐴 [∅ ∈ 𝐴 ∧ ∀𝑎 (𝑎 ∈ 𝐴 ⇒ 𝑎+ ∈ 𝐴)].







∃𝐴 [∅ ∈ 𝐴 ∧ ∀𝑎 (𝑎 ∈ 𝐴 ⇒ 𝑎+ ∈ 𝐴)].







∃𝐴 [∅ ∈ 𝐴 ∧ ∀𝑎 (𝑎 ∈ 𝐴 ⇒ 𝑎+ ∈ 𝐴)].


Natural NumbersDefinitionA natural number is a set that belongs to every inductive set.

Theorem 4A (Enderton 1977)There is a set whose members are exactly the natural numbers.

DefinitionThe set of all natural numbers, denoted by 𝜔, is defined by

𝑥 ∈ 𝜔 ⇔ 𝑥 is a natural number.

Theorem 4B (Enderton 1977)The set 𝜔 is inductive, and is a subset of every other inductive set.




















Induction Principle for Natural Numbers

Induction principle for 𝜔 (one version)Any inductive subset of 𝜔 coincides with 𝜔 (Enderton 1977, p. 69).

Induction principle for 𝜔 (other version)Let 𝑃 (𝑥) be a property. Assume that

i) 𝑃 (0) holdsii) For all 𝑛 ∈ 𝜔, 𝑃(𝑛) implies 𝑃(𝑛+)

Then 𝑃 holds for all natural numbers 𝑛.

Proof.“This is an immediate consequence of our definition of 𝑤. The assump-tions i) and ii) simple say that the set 𝐴 = {𝑛 ∈ 𝜔 ∣ 𝑃(𝑛)} is induct-ive. 𝜔 ⊆ 𝐴 follows.” (Hrbacek and Jech 1999, p. 42)


Induction Principle for Natural Numbers

Induction principle for 𝜔 (one version)Any inductive subset of 𝜔 coincides with 𝜔 (Enderton 1977, p. 69).

Induction principle for 𝜔 (other version)Let 𝑃(𝑥) be a property. Assume that

i) 𝑃(0) holdsii) For all 𝑛 ∈ 𝜔, 𝑃(𝑛) implies 𝑃(𝑛+)

Then 𝑃 holds for all natural numbers 𝑛.

Proof.“This is an immediate consequence of our definition of 𝑤. The assump-tions i) and ii) simple say that the set 𝐴 = {𝑛 ∈ 𝜔 ∣ 𝑃(𝑛)} is induct-ive. 𝜔 ⊆ 𝐴 follows.” (Hrbacek and Jech 1999, p. 42)


Recursion on Natural Numbers

Recursion theorem on 𝜔 (Enderton 1977, p. 73)Let 𝐴 be a set, 𝑎 ∈ 𝐴 and 𝐹 ∶ 𝐴 → 𝐴. Then there exists a unique functionℎ ∶ 𝜔 → 𝐴 such that

ℎ(0) = 𝑎,and for every 𝑛 in 𝜔

ℎ(𝑛+) = 𝐹(ℎ(𝑛)).


Defining Natural Numbers as SetsRemarkSo far, we defined natural numbers on terms of sets. A different point ofview is stated by some authors (see, for example, Benacerraf (1965)).


Induction as Foundations“Thus inductive definibility is anotion intermediate in strengthbetween predicate and fullyimpredicative definability.” (Aczel1977, p. 780)

It would be interesting to formulate acoherent conceptual framework thatmade induction the principal notion.There are suggestions of this in theliterature, but the possibility has notyet been fully explored.” Ibid.


Cardinal Numbers

EquinumerosityRemarkA one-to-one function from 𝐴 onto 𝐵 is called a one-to-one correspond-ence between 𝐴 and 𝐵.

DefinitionA set 𝐴 is equinumerous to a set 𝐵, denoted 𝐴 ≈ 𝐵, iff there is a one-to-one correspondence between 𝐴 and 𝐵.


Cardinal Numbers 102/144









Equinumerosity

“The possibility that whole and partmay have the same number of termsis, it must be confessed, shocking tocommon-sense.” (Russell 1903,p. 358)


Equinumerosity

Theorem 6B(a) (Enderton 1977)The set 𝜔 is not equinumerous to the set ℝ of real numbers.

Proof.Let’s suppose 𝜔 ≈ ℝ, that is, there is an one-to-one correspondence𝑓 ∶ 𝜔 → ℝ such that∗

𝑓(0) = 236.001 … ,𝑓(1) = −7.777 … ,𝑓(2) = 3.1415 … ,

⋮

(continues in the next slide)∗“We assume that a decimal expansion does not contain only the digit 9 from some

place on, so each real number has a unique decimal expansion.” (Hrbacek and Jech1999, Theorem 6.1, p. 90)


Equinumerosity

Proof (cont).Let 𝑥 = 0.𝑑1𝑑2𝑑3 … ∈ ℝ, where

𝑑𝑛+1 = {4, if 𝑓(𝑛) ≠ 4;5, if 𝑓(𝑛) = 4.

The number 𝑥 doesn’t belong to the above enumeration. Therefore, ℝ isnon-enumerable.


On Refutations of Cantor’s Diagonal Argument“I dedicate this essay to thetwo-dozen-odd people whoserefutations of Cantor’s diagonalargument have come to me either asreferee or as editor in the last twentyyears or so... A few years ago itoccurred to me to wonder why somany people devote so much energyto refuting this harmless littleargument—what had it done tomake them angry with it?... Thesepages report the results.” (Hodges1998, p. 1)


Finite SetsDefinitionA set is finite iff it is equinumerous to some natural number. Otherwise itis infinite.

Corollary 6C (Enderton 1977)No finite set is equinumerous to a proper subset of itself.

Corollary 6D (Enderton 1977)1. Any set equinumerous to a proper subset of itself is infinite.2. The set 𝜔 is infinite.








Corollary 6D (Enderton 1977)1. Any set equinumerous to a proper subset of itself is infinite.

2. The set 𝜔 is infinite.






The Continuum Hypothesis

The continuum hypothesis (CH)There is no a set whose cardinality is strictly between the cardinality of theset of the natural numbers and the cardinality of the set of real numbers,that is,

2ℵ0 = ℵ1.CH could not be disproved (Gödel 1938) nor proved (Cohen 1963) in ZFC,that is, CH is independent of ZFC set theory.


Orderings and Ordinals

Well OrderingsDefinitionA well ordering on 𝐴 is a linear ordering on 𝐴 with the further propertythat every nonempty subset of 𝐴 has a least element.

DefinitionA structure is a pair ⟨𝐴, 𝑅⟩ consisting of a set 𝐴 and a binary relation 𝑅on 𝐴.

Orderings and Ordinals 115/144

Well OrderingsDefinitionA well ordering on 𝐴 is a linear ordering on 𝐴 with the further propertythat every nonempty subset of 𝐴 has a least element.

DefinitionA structure is a pair ⟨𝐴, 𝑅⟩ consisting of a set 𝐴 and a binary relation 𝑅on 𝐴.


Transfinite Induction PrincipleDefinitionLet < be some sort of ordering on 𝐴 and 𝑡 ∈ 𝐴. The set

seg 𝑡 = {𝑥 ∣ 𝑥 < 𝑡}

is called the initial segment up to 𝑡.

Transfinite induction principleLet ⟨𝐴, <⟩ be a well-ordered structure and assume that 𝐵 is a subset of 𝐴with the special property that for every 𝑡 in 𝐴,

seg 𝑡 ⊆ 𝐵 ⇒ 𝑡 ∈ 𝐵.

Then 𝐵 coincides with 𝐴.


Transfinite Induction PrincipleDefinitionLet < be some sort of ordering on 𝐴 and 𝑡 ∈ 𝐴. The set

seg 𝑡 = {𝑥 ∣ 𝑥 < 𝑡}

is called the initial segment up to 𝑡.

Transfinite induction principleLet ⟨𝐴, <⟩ be a well-ordered structure and assume that 𝐵 is a subset of 𝐴with the special property that for every 𝑡 in 𝐴,

seg 𝑡 ⊆ 𝐵 ⇒ 𝑡 ∈ 𝐵.

Then 𝐵 coincides with 𝐴.


IsomorphismsDefinitionLet ⟨𝐴, 𝑅⟩ and ⟨𝐵, 𝑆⟩ be two structures. An isomorphism from ⟨𝐴, 𝑅⟩onto ⟨𝐵, 𝑆⟩ is a one-to-one function 𝑓 from 𝐴 onto 𝐵 such that for all𝑥, 𝑦 ∈ 𝐴

𝑥𝑅𝑦 iff 𝑓(𝑥)𝑆𝑓(𝑦).


Ordinal NumbersDefinitionLet < be a well-ordering on 𝐴. The ordinal number of ⟨𝐴, <⟩ is its 𝜀-image. An ordinal number is a set that is the ordinal number of somewell-ordered structure.


Ordinal NumbersDefinitionA set 𝐴 is well-ordered by epsilon iff the relation

∈𝐴= {⟨𝑥, 𝑦⟩ ∈ 𝐴 × 𝐴 ∣ 𝑥 ∈ 𝑦}

is a well ordering on 𝐴.

Remark (other definition of ordinal number)A set 𝐴 is an ordinal number iff (Hrbacek and Jech 1999, p. 107):

1. The set is transitive.2. The set is well-ordered by ∈𝐴.


Ordinal NumbersDefinitionA set 𝐴 is well-ordered by epsilon iff the relation

∈𝐴= {⟨𝑥, 𝑦⟩ ∈ 𝐴 × 𝐴 ∣ 𝑥 ∈ 𝑦}

is a well ordering on 𝐴.

Remark (other definition of ordinal number)A set 𝐴 is an ordinal number iff (Hrbacek and Jech 1999, p. 107):

1. The set is transitive.2. The set is well-ordered by ∈𝐴.


Ordinal NumbersRemarkThe class of ordinal numbers, denoted by On, can be defined by (Rubin1967, pp. 175-176):

0 ∈ On,𝛼 ∈ On ⇒ 𝛼+ ∈ On,

𝐴 ⊆ On ⇒ ⋃ 𝐴 ∈ On.

See also Enderton (1977, Corollary 7N).

Burali-Forti theoremThere is no set to which every ordinal number belongs.



0 ∈ On,𝛼 ∈ On ⇒ 𝛼+ ∈ On,

𝐴 ⊆ On ⇒ ⋃ 𝐴 ∈ On.





0 ∈ On,𝛼 ∈ On ⇒ 𝛼+ ∈ On,

𝐴 ⊆ On ⇒ ⋃ 𝐴 ∈ On.




Cardinal NumbersDefinitionLet 𝐴 be a set. The cardinal number of 𝐴, denoted card 𝐴, is the leastordinal equinumerous to 𝐴.

DefinitionAn ordinal number is an initial ordinal iff it is not equinumerous to anysmaller ordinal number.

RemarkCardinal numbers and initial ordinals are the same numbers.










Ordinals and Order Types

Ordinal AdditionDefinitionLet 𝛼 and 𝛽 be ordinals. We define their addition by transfinite recursionon 𝛽:

𝛼 + 0 = 𝛼,

𝛼 + 𝛽+ = (𝛼 + 𝛽)+,

𝛼 + 𝛽 = sup{𝛼 + 𝜆 ∣ 𝜆 < 𝛽}= ⋃

𝜆<𝛽(𝛼 + 𝜆), for all limit ordinal 𝛽.

Ordinals and Order Types 130/144

Ordinal AdditionExamples𝜔 = 1 + 𝜔, 1 + 𝜔 ≠ 𝜔 + 1 and 𝜔 + 1 ≠ 𝜔 + 𝜔.

0123

⋮

(a) 𝜔⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨2, 1⟩

⋮

(b) 1 + 𝜔⟨0, 0⟩⟨1, 0⟩⟨2, 0⟩

⋮⟨0, 1⟩

(c) 𝜔 + 1⟨0, 0⟩⟨1, 0⟩⟨2, 0⟩

⋮⟨0, 1⟩⟨1, 1⟩⟨2, 1⟩

⋮

(d) 𝜔 + 𝜔

Ordinals and Order Types 131/144

List of Axioms

Extensionality axiom: If two sets have exactly the same members,then they are equal, that is,

∀𝐴 ∀𝐵 [∀𝑥 (𝑥 ∈ 𝐴 ⇔ 𝑥 ∈ 𝐵) ⇒ 𝐴 = 𝐵].

Empty (existence) axiom: There is a set having no members,that is,

∃𝐵 ∀𝑥 (𝑥 ∉ 𝐵).

Pairing axiom: For any sets 𝑢 and 𝑣, there is a set having asmembers just 𝑢 and 𝑣, that is,

∀𝑎 ∀𝑏 ∃𝐶 ∀𝑥 (𝑥 ∈ 𝐶 ⇔ 𝑥 = 𝑎 ∨ 𝑥 = 𝑏).

List of Axioms 132/144

List of Axioms

Union axiom (first version): For any sets 𝑎 and 𝑏, there is a setwhose members are those sets belonging either to 𝑎 or to 𝑏 (or both),that is,

∀𝑎 ∀𝑏 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑎 ∨ 𝑥 ∈ 𝑏).

Union axiom (final version): For any set 𝐴, there exists a set 𝐵whose elements are exactly the members of the members of 𝐴, that is,

∀𝐴 ∃𝐵 ∀𝑥 [𝑥 ∈ 𝐵 ⇔ ∃𝑏 (𝑏 ∈ 𝐴 ∧ 𝑥 ∈ 𝑏)].

Power set axiom: For any set 𝑎, there is a set whose members areexactly the subsets of 𝑎, that is,

∀𝑎 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ⊆ 𝑎).


List of Axioms

Subset axiom scheme (axiom scheme of comprehension orseparation): For each formula 𝜑, not containing 𝐵, the following is anaxiom:

∀𝑡1 ⋯ ∀𝑡𝑘 ∀𝑐 ∃𝐵 ∀𝑥 (𝑥 ∈ 𝐵 ⇔ 𝑥 ∈ 𝑐 ∧ 𝜑).

Infinity axiom: There exists an inductive set, that is,

∃𝐴 [∅ ∈ 𝐴 ∧ ∀𝑎 (𝑎 ∈ 𝐴 ⇒ 𝑎+ ∈ 𝐴)].

Axiom of choice (a version): For any relation 𝑅 there is a function𝐹 ⊆ 𝑅 with dom 𝐹 = dom 𝑅.


List of Axioms

Regularity (foundation) axiom: All sets are well-founded, that is,

∀𝐴 [𝐴 ≠ ∅ ⇒ ∃𝑚 (𝑚 ∈ 𝐴 ∧ 𝑚 ∩ 𝐴 = ∅)].


Appendix: Formal Proofs

Proofs by Contradiction and Proofs of Negations

Proof by contradiction(or reductio ad absurdum)

[¬𝛽]⋮

⊥𝛽

Proof of negation (Bauer 2017)

[𝛽]⋮⊥¬𝛽

Justifications

[¬𝛽]⋮

⊥ (conditional proof)¬𝛽 → ⊥(¬𝛼 def= 𝛼 → ⊥)¬¬𝛽 (⊢ ¬¬𝛼 → 𝛼)𝛽

[𝛽]⋮

⊥ (conditional proof)𝛽 → ⊥(¬𝛼 def= 𝛼 → ⊥)¬𝛽

Appendix: Formal Proofs 137/144

Proofs by Contradiction and Proofs of Negations

Proof by contradiction(or reductio ad absurdum)

[¬𝛽]⋮

⊥𝛽

Proof of negation (Bauer 2017)

[𝛽]⋮⊥¬𝛽

Justifications

[¬𝛽]⋮

⊥ (conditional proof)¬𝛽 → ⊥(¬𝛼 def= 𝛼 → ⊥)¬¬𝛽 (⊢ ¬¬𝛼 → 𝛼)𝛽

[𝛽]⋮

⊥ (conditional proof)𝛽 → ⊥(¬𝛼 def= 𝛼 → ⊥)¬𝛽


Natural Deduction: Derivations Rules𝜑 𝜓 ∧I𝜑 ∧ 𝜓

𝜑 ∧ 𝜓 ∧E𝜑𝜑 ∧ 𝜓 ∧E𝜓

[𝜑]⋮𝜓 →I𝜑 → 𝜓

𝜑 𝜑 → 𝜓 →E𝜓

𝜑 ∨I𝜑 ∨ 𝜓𝜓 ∨I𝜑 ∨ 𝜓 𝜑 ∨ 𝜓

[𝜑]⋮𝜎

[𝜓]⋮𝜎 ∨E𝜎

⊥ ⊥E𝜑

[¬𝜑]⋮⊥ RAA𝜑


Natural Deduction: Derivations RulesRemarks

Abbreviations: ¬𝜑 ∶= 𝜑 → ⊥.In the application of the →I rule, we may discharge zero, one, or moreoccurrences of the assumption.References: See, for example, (van Dalen 2013).


Examples

Example (van Dalen (2013), example p. 49)Proof of ⊢ 𝜑 ∨ ¬𝜑.

[𝜑]1∨I𝜑 ∨ ¬𝜑 [¬(𝜑 ∨ ¬𝜑)]2

→E⊥

→I1¬𝜑∨I𝜑 ∨ ¬𝜑 [¬(𝜑 ∨ ¬𝜑)]2

→E⊥

RAA2𝜑 ∨ ¬𝜑


References

Aczel, Peter (1977). An Introduction to Inductive Definitions. In: Handbookof Mathematical Logic. Ed. by Barwise, Jon. Vol. 90. Studies in Logic andthe Foundations of Mathematics. Elsevier. Chap. C.7. doi:10.1016/S0049-237X(08)71120-0 (cit. on p. 100).Bauer, Andrej (2017). Five States of Accepting Constructive Mathematics.Bull. Amer. Math. Soc. 54.3, pp. 481–498. doi: 10.1090/bull/1556(cit. on pp. 137, 138).Benacerraf, Paul (1965). What Numbers Could not Be. The PhilosophicalReview 74.1, pp. 47–73. doi: 10.2307/2183530 (cit. on p. 99).Cantor, Georg (1895). Beiträge zur Begründung der transfinitenMengenlehre. Mathematische Annalen 46.4, pp. 481–512. doi:10.1007/BF02124929 (cit. on pp. 7, 8).Cohen, Paul J. (1963). The Independence of the Continuum Hypothesis.Proceedings of the National Academy of Sciences of the United States ofAmerica 50.6, pp. 1143–1148. url:http://www.pnas.org/content/50/6/1143.full.pdf (cit. on p. 113).

https://doi.org/10.1016/S0049-237X(08)71120-0

https://doi.org/10.1090/bull/1556

https://doi.org/10.2307/2183530

https://doi.org/10.1007/BF02124929

http://www.pnas.org/content/50/6/1143.full.pdf

ReferencesEnderton, Herbert B. (1977). Elements of Set Theory. Academic Press(cit. on pp. 19, 20, 25, 41, 42, 45–48, 55, 56, 58, 59, 67–70, 92–98, 106,109–112, 123–125).Gödel, Kurt (1938). The Consistency of the Axiom of Choice and of theGeneralized Continuum-Hypothesis. Proceedings of the National Academyof Sciences of the United States of America 24.12, pp. 556–557. doi:10.1073/pnas.24.12.556 (cit. on p. 113).Hamilton, A. G. (1978). Logic for Mathematicians. Revised edition.Cambridge University Press (cit. on pp. 27, 28).Hodges, Wilfrid (1998). An Editor Recalls some Hopeless Papers. TheBulletin of Symbolic Logic 4.1, pp. 1–16. doi: 10.2307/421003 (cit. onp. 108).Hrbacek, Karel and Jech, Thomas (1999). Introduction to Set Theory.Third Edition, Revised and Expanded. Marcel Dekker (cit. on pp. 7, 8,11–13, 25, 86, 96, 97, 106, 121, 122).

https://doi.org/10.1073/pnas.24.12.556

https://doi.org/10.2307/421003

ReferencesMartin-Löf, Per (2006). 100 Years of Zermelo’s Axiom of Choice: What wasthe Problem with It? The Computer Journal 49.3, pp. 345–350. doi:10.1093/comjnl/bxh162 (cit. on pp. 60, 61).Mendelson, Elliott (1997). Introduction to Mathematical Logic. 4th ed.Chapman & Hall (cit. on pp. 27, 28).Moore, Gregory H. (1982). Zermelo’s Axiom of Choice. Its Origins,Development, and Influence. Springer-Verlag (cit. on pp. 11–13).Rubin, Jean E. (1967). Set Theory for the Mathematician. Holden-Day(cit. on pp. 123–125).Russell, Bertrand (1903). The Principles of Mathematics. W. W. Norton &Company, Inc (cit. on p. 105).van Dalen, Dirk (2013). Logic and Structure. 5th ed. Springer (cit. onpp. 140, 141).van Heijenoort, Jean (1967). From Frege to Gödel: A Source Book inMathematical Logic, 1879-1931. Harvard University Press (cit. on p. 17).

https://doi.org/10.1093/comjnl/bxh162

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