Logic and the set theory - Lecture 19: The set theorymathsci.kaist.ac.kr/~schoi/Logiclec19.pdf ·...

Logic and the set theoryLecture 19: The set theory

S. Choi

Department of Mathematical ScienceKAIST, Daejeon, South Korea

Fall semester, 2012

S. Choi (KAIST) Logic and set theory November 20, 2012 1 / 24

Introduction

About this lecture

Axioms of the set theory

I Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.htmland the moodle page http://moodle.kaist.ac.kr

Grading and so on in the moodle. Ask questions in moodle.

Introduction

About this lecture

Axioms of the set theoryI Axiom of extension

I Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specification

I Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null set

I Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairing

I Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unions

I Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power sets

I Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinity

I Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularity

I Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Axioms of the set theoryI Axiom of extensionI Axiom of specificationI Axiom of null setI Axiom of pairingI Axiom of unionsI Axiom of power setsI Axiom of infinityI Axiom of regularityI Axiom of choice

Russell’s paradox

Natural numbers

Introduction

About this lecture

Russell’s paradox

Natural numbers

Introduction

About this lecture

Russell’s paradox

Natural numbers

Introduction

About this lecture

Russell’s paradox

Natural numbers

Introduction

About this lecture

Russell’s paradox

Natural numbers

Introduction

Some helpful references

Sets, Logic and Categories, Peter J. Cameron, Springer. Read Chapters 3,4,5.

http://plato.stanford.edu/contents.html has much resource.

Introduction to set theory, Hrbacek and Jech, CRC Press. (Chapter 3 (3.2, 3.3))

Introduction to mathematical logic: set theory, computable functions, model theory,Malitz, J. Springer

Sets for mathematics, F.W. Lawvere, R. Rosebrugh, Cambridge

Introduction

Naive set theory

Naive set theory (Zermelo-Fraenkel, ZFC)

Sets{}.

∈, ⊂, ∅These satisfy certain axioms.

The meaning of the symbols are not given per say... You can read ∈ as "is anancester of ".

The axioms are in fact temporary ones until we find better ones.

The main reason that they exist is to aid in the proof and to follow the classicallogic, and finally to avoid possible self-contradictions such as Russell’s.

The set theory can be characterized within the category theory.

Naive set theory

Sets{}.∈, ⊂, ∅

These satisfy certain axioms.

Naive set theory

Sets{}.∈, ⊂, ∅These satisfy certain axioms.

Naive set theory

Axiom of extension

Two sets are equal if and only if they have the same elements.

A = B and A 6= B.

A ⊂ B is defined as ∀x(x ∈ A→ x ∈ B).

Reflexivity A ⊂ A.

Transitivity: A ⊂ B, B ⊂ C Then A ⊂ C.

Antisymmetry: A ⊂ B and B ⊂ A. Then A = B.

Naive set theory

Axiom of extension

A = B and A 6= B.

Naive set theory

Axiom of extension

A = B and A 6= B.

Naive set theory

Axiom of extension

A = B and A 6= B.

Naive set theory

Axiom of extension

A = B and A 6= B.

Naive set theory

Axiom of extension

A = B and A 6= B.

Naive set theory

Axiom of specification

To every set and to any condition S(x) there corresponds a set B whose elementsare exactly those of A satisfying S(x).

A the set of all men. {x ∈ A|x is married .}.

Naive set theory

Axiom of specification

To every set and to any condition S(x) there corresponds a set B whose elementsare exactly those of A satisfying S(x).

A the set of all men. {x ∈ A|x is married .}.

Naive set theory

Axiom of a null-set

A set with no element exists {} or ∅.

∅ ⊂ A for any set A.

Naive set theory

Axiom of a null-set

A set with no element exists {} or ∅.∅ ⊂ A for any set A.

Naive set theory

Axiom of pairing

For any two sets, there exists a set that they both belong to.

Sets X ,Y . There exists Z 3 X ,Y .

There exists {a, b}. (To see this use the first item to get a set A and form{x ∈ A|x = a, x = b}.Singleton {a} exists.

Example: {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}, ....

Naive set theory

Axiom of pairing

Example: {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}, ....

Naive set theory

Axiom of pairing

There exists {a, b}. (To see this use the first item to get a set A and form{x ∈ A|x = a, x = b}.

Singleton {a} exists.

Example: {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}, ....

Naive set theory

Axiom of pairing

Example: {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}, ....

Naive set theory

Axiom of pairing

Example: {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}, ....

Naive set theory

Axiom of union

Given a collection of sets, there exists a set U that contains all the elements thatbelong to at least one set in the given collection C.

U may be too general.

We specify⋃

X∈C X := {x ∈ U|∃X ∈ C(x ∈ X )}.Example:

⋃{X |X ∈ ∅} = ∅.⋃

{A1,A2, ..,An} = A1 ∪ A2 ∪ · · · ∪ An.

Naive set theory

Axiom of union

Given a collection of sets, there exists a set U that contains all the elements thatbelong to at least one set in the given collection C.U may be too general.

We specify⋃

⋃{X |X ∈ ∅} = ∅.⋃

{A1,A2, ..,An} = A1 ∪ A2 ∪ · · · ∪ An.

Naive set theory

Axiom of union

We specify⋃

X∈C X := {x ∈ U|∃X ∈ C(x ∈ X )}.

Example:⋃{X |X ∈ ∅} = ∅.⋃

{A1,A2, ..,An} = A1 ∪ A2 ∪ · · · ∪ An.

Naive set theory

Axiom of union

We specify⋃

⋃{X |X ∈ ∅} = ∅.

⋃{A1,A2, ..,An} = A1 ∪ A2 ∪ · · · ∪ An.

Naive set theory

Axiom of union

We specify⋃

⋃{X |X ∈ ∅} = ∅.⋃

{A1,A2, ..,An} = A1 ∪ A2 ∪ · · · ∪ An.

Naive set theory

Axiom of union

⋃{X |X ∈ {A}} = A.

A ∪ ∅ = A, A ∪ B = B ∪ A.

Prove for example {a} ∪ {b} ∪ {c} = {a, b, c}.(A ∪ B) ∪ C = A ∪ (B ∪ C).

A ∪ A = A.

A ⊂ B ↔ A ∪ B = B.

Naive set theory

Axiom of union

⋃{X |X ∈ {A}} = A.

A ∪ ∅ = A, A ∪ B = B ∪ A.

A ∪ A = A.

A ⊂ B ↔ A ∪ B = B.

Naive set theory

Axiom of union

⋃{X |X ∈ {A}} = A.

A ∪ ∅ = A, A ∪ B = B ∪ A.

Prove for example {a} ∪ {b} ∪ {c} = {a, b, c}.

(A ∪ B) ∪ C = A ∪ (B ∪ C).

A ∪ A = A.

A ⊂ B ↔ A ∪ B = B.

Naive set theory

Axiom of union

⋃{X |X ∈ {A}} = A.

A ∪ ∅ = A, A ∪ B = B ∪ A.

A ∪ A = A.

A ⊂ B ↔ A ∪ B = B.

Naive set theory

Axiom of union

⋃{X |X ∈ {A}} = A.

A ∪ ∅ = A, A ∪ B = B ∪ A.

A ∪ A = A.

A ⊂ B ↔ A ∪ B = B.

Naive set theory

Axiom of union

⋃{X |X ∈ {A}} = A.

A ∪ ∅ = A, A ∪ B = B ∪ A.

A ∪ A = A.

A ⊂ B ↔ A ∪ B = B.

Naive set theory

Intersection

For every collection C, other than ∅, there exists a set V such that x ∈ V if andonly if x ∈ X for every X ∈ C.

Let A be an element of C. Then V = {x ∈ A|∀X ∈ Cx ∈ X}.V = {x |∀X ∈ Cx ∈ X}.This is uniquely defined by the axiom of extension.

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Defintion of A1 ∩ A2 ∩ · · · ∩ An.

Let ∅ be a family of subsets of E . Then⋂

X∈∅ X = E . (For no X , x ∈ X is false.)

Not specifying E gets you into trouble.

Naive set theory

Intersection

For every collection C, other than ∅, there exists a set V such that x ∈ V if andonly if x ∈ X for every X ∈ C.Let A be an element of C. Then V = {x ∈ A|∀X ∈ Cx ∈ X}.

V = {x |∀X ∈ Cx ∈ X}.This is uniquely defined by the axiom of extension.

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Intersection

For every collection C, other than ∅, there exists a set V such that x ∈ V if andonly if x ∈ X for every X ∈ C.Let A be an element of C. Then V = {x ∈ A|∀X ∈ Cx ∈ X}.V = {x |∀X ∈ Cx ∈ X}.

This is uniquely defined by the axiom of extension.

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Intersection

For every collection C, other than ∅, there exists a set V such that x ∈ V if andonly if x ∈ X for every X ∈ C.Let A be an element of C. Then V = {x ∈ A|∀X ∈ Cx ∈ X}.V = {x |∀X ∈ Cx ∈ X}.This is uniquely defined by the axiom of extension.

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Intersection

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Intersection

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Intersection

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Intersection

Denote by⋂C or

⋂{X : X ∈ C} or

⋂X∈C X .

Naive set theory

Distributivity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

In fact A ∩⋃C =

⋃C1 where C1 = {A ∩ B|A ∩ B 6= ∅ ∧ B ∈ C}

A ∪⋂C =

⋂C2 where C2 = {A ∪ B|B ∈ C}

Naive set theory

Distributivity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

In fact A ∩⋃C =

⋃C1 where C1 = {A ∩ B|A ∩ B 6= ∅ ∧ B ∈ C}

A ∪⋂C =

⋂C2 where C2 = {A ∪ B|B ∈ C}

Naive set theory

Distributivity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

In fact A ∩⋃C =

⋃C1 where C1 = {A ∩ B|A ∩ B 6= ∅ ∧ B ∈ C}

A ∪⋂C =

⋂C2 where C2 = {A ∪ B|B ∈ C}

Naive set theory

Distributivity

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

In fact A ∩⋃C =

⋃C1 where C1 = {A ∩ B|A ∩ B 6= ∅ ∧ B ∈ C}

A ∪⋂C =

⋂C2 where C2 = {A ∪ B|B ∈ C}

Naive set theory

Complements

A− B = {x ∈ A|x /∈ B}.

E . The set containing all sets here.

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A ⊂ B if and only if B′ ⊂ A′.

(A ∪ B)′ = A′ ∩ B′, (A ∩ B)′ = A′ ∪ B′. De Morgan’s law

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A− B = {x ∈ A|x /∈ B}.E . The set containing all sets here.

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.

A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

Complements

A′ := E − A.

(A′)′ = A.

∅′ = E ,E ′ = ∅.A ∩ A′ = ∅ and A ∪ A′ = E .

A4B = (A− B) ∪ (B − A). (A + B in NS)

Naive set theory

The axiom of the power set

For every set, there is a collection of sets that contains among its elements all thesubsets of the given set.

For every x , there is y such that for all z, z ∈ y iff z ⊂ x .

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

X , P(X ) the power set of X

P(X ) = {x |x ⊂ X}.P(∅) = {∅}.P({a, b}) =?

Naive set theory

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

P(X ) = {x |x ⊂ X}.P(∅) = {∅}.P({a, b}) =?

Naive set theory

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

P(X ) = {x |x ⊂ X}.P(∅) = {∅}.P({a, b}) =?

Naive set theory

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

P(X ) = {x |x ⊂ X}.P(∅) = {∅}.P({a, b}) =?

Naive set theory

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

P(X ) = {x |x ⊂ X}.

P(∅) = {∅}.P({a, b}) =?

Naive set theory

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

P(X ) = {x |x ⊂ X}.P(∅) = {∅}.

P({a, b}) =?

Naive set theory

∀x∃y∀z(z ∈ y ↔ z ⊂ x).

P(X ) = {x |x ⊂ X}.P(∅) = {∅}.P({a, b}) =?

Naive set theory

The power set

P(E) ∩ P(F ) = P(E ∩ F ), P(E) ∪ P(F ) ⊂ P(E ∪ F ).

⋂X∈C P(X ) = P(

⋂X∈C X ).⋃

X∈C P(X ) ⊂ P(⋃

X∈C X ).⋂X∈P(E) X = ∅.

E ⊂ F → P(E) ⊂ P(F ).

E =⋃

Naive set theory

The power set

P(E) ∩ P(F ) = P(E ∩ F ), P(E) ∪ P(F ) ⊂ P(E ∪ F ).⋂X∈C P(X ) = P(

⋂X∈C X ).

⋃X∈C P(X ) ⊂ P(

⋃X∈C X ).⋂

X∈P(E) X = ∅.E ⊂ F → P(E) ⊂ P(F ).

E =⋃

Naive set theory

The power set

⋂X∈C X ).⋃

X∈C P(X ) ⊂ P(⋃

X∈C X ).

⋂X∈P(E) X = ∅.

E ⊂ F → P(E) ⊂ P(F ).

E =⋃

Naive set theory

The power set

⋂X∈C X ).⋃

X∈C P(X ) ⊂ P(⋃

X∈C X ).⋂X∈P(E) X = ∅.

E ⊂ F → P(E) ⊂ P(F ).

E =⋃

Naive set theory

The power set

⋂X∈C X ).⋃

X∈C P(X ) ⊂ P(⋃

X∈C X ).⋂X∈P(E) X = ∅.

E ⊂ F → P(E) ⊂ P(F ).

E =⋃

Naive set theory

The power set

⋂X∈C X ).⋃

X∈C P(X ) ⊂ P(⋃

X∈C X ).⋂X∈P(E) X = ∅.

E ⊂ F → P(E) ⊂ P(F ).

E =⋃

Naive set theory

Question

What type of sets can you construct with the axioms given? Give us examples...

{∅, {∅, {∅}}}, ...

Can we get an infinite set now?

Naive set theory

Question

{∅, {∅, {∅}}}, ...

Naive set theory

Question

{∅, {∅, {∅}}}, ...

Naive set theory

Axiom of infinity

There is a set X such that φ ∈ X and whenever y ∈ X , then y ∪ {y} ∈ X .

Numbers x+ = x ∪ {x}.0 := ∅, 1 = 0+ = {0} = {∅}, 2 = 1+ = {0, 1} = {∅, {∅}},3 = 2+ = {0, 1, 2} = {0, 1, {0, 1}} = {∅, {∅}, {∅, {∅}}}.4 = 3+ =???.

Naive set theory

Axiom of infinity

Numbers x+ = x ∪ {x}.

0 := ∅, 1 = 0+ = {0} = {∅}, 2 = 1+ = {0, 1} = {∅, {∅}},3 = 2+ = {0, 1, 2} = {0, 1, {0, 1}} = {∅, {∅}, {∅, {∅}}}.4 = 3+ =???.

Naive set theory

Axiom of infinity

Numbers x+ = x ∪ {x}.0 := ∅, 1 = 0+ = {0} = {∅}, 2 = 1+ = {0, 1} = {∅, {∅}},

3 = 2+ = {0, 1, 2} = {0, 1, {0, 1}} = {∅, {∅}, {∅, {∅}}}.4 = 3+ =???.

Naive set theory

Axiom of infinity

Numbers x+ = x ∪ {x}.0 := ∅, 1 = 0+ = {0} = {∅}, 2 = 1+ = {0, 1} = {∅, {∅}},3 = 2+ = {0, 1, 2} = {0, 1, {0, 1}} = {∅, {∅}, {∅, {∅}}}.

4 = 3+ =???.

Naive set theory

Axiom of infinity

Numbers x+ = x ∪ {x}.0 := ∅, 1 = 0+ = {0} = {∅}, 2 = 1+ = {0, 1} = {∅, {∅}},3 = 2+ = {0, 1, 2} = {0, 1, {0, 1}} = {∅, {∅}, {∅, {∅}}}.4 = 3+ =???.

Naive set theory

Definition of N or ω.

Temporary definition: A set A is a successor set if 0 ∈ A and y+ ∈ A whenevery ∈ A.

The axiom of infinity says a successor set exists, say A.

The intersection of a family of successor sets is a successor set. Proof: ?

Define ω as the intersection of all collection of successor sets in A.

Then ω is a subset of every successor set:

Proof: Let B be a successor set. Then ω ⊂ A ∩ B since ?. Thus, ω ⊂ B.

ω is uniquely defined:

Proof: ω exists. Suppose ω′ is another...

A natural number is an element of ω: 0, 1, 2, ....

Naive set theory

The axiom of regularity (foundation)

Every nonempty set has an ∈-least member. That is, if there is some y ∈ x , thenthere exists z ∈ x for which there is no w ∈ z ∩ x . (There is no element of x that isan element of z).

Consequences:

(1) No nonempty set can be a member of itself. No A = {A, ...}...(2) If A,B are both nonempty sets, then it is not possible that both A ∈ B andB ∈ A are true.

Naive set theory

Consequences:

Naive set theory

Consequences:

(1) No nonempty set can be a member of itself. No A = {A, ...}...

(2) If A,B are both nonempty sets, then it is not possible that both A ∈ B andB ∈ A are true.

Naive set theory

Consequences:

Naive set theory

The proofs

(1) Suppose that ∃A such that A ∈ A, and A is not empty, then {A} would be a set.A is the only element.

However, in {A} there is no z as above.

(2) Suppose there exists nonempty sets A and B with A ∈ B and B ∈ A. Then{A,B} is a set.

B ∈ A ∩ {A,B} since B ∈ A.

A ∈ B ∩ {A,B} since A ∈ B.

Thus there is no z for {A,B} since we can keep finding one less than any element.

Naive set theory

The proofs

Naive set theory

The proofs

Naive set theory

The proofs

Naive set theory

The proofs

Naive set theory

The proofs

Naive set theory

The proofs

Russell’s paradox

Let A be a set.

Define B = {x ∈ A : ¬(x ∈ x)}:y ∈ B ↔ y ∈ A ∧ y /∈ y .

Is B ∈ A?If B ∈ A, then

I (i) B 6∈ B: Then B ∈ A and B 6∈ B imply B ∈ B. Contradiction.I (ii) B ∈ B: Then B ∈ A implies B 6∈ B. Contr.

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

Define B = {x ∈ A : ¬(x ∈ x)}:

y ∈ B ↔ y ∈ A ∧ y /∈ y .

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

Is B ∈ A?

If B ∈ A, then

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

I (i) B 6∈ B: Then B ∈ A and B 6∈ B imply B ∈ B. Contradiction.

I (ii) B ∈ B: Then B ∈ A implies B 6∈ B. Contr.

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

Thus, B 6∈ A.

Russell’s paradox

Let A be a set.

Thus, B 6∈ A.

Russell’s paradox

Let U be the “set” of all sets.

A set x is normal if x 6∈ x . the “set” of all normal sets.

Let A = {x ∈ U : ¬(x ∈ x)}. The “set” of all normal sets.

By above A 6∈ U. But U is the “set” of all sets: a contradiction.

Also, A 6∈ A if and only if A ∈ A by definition: a contradiction.

Hence A is not a set. (a collection)

There is no ‘universal set’ U.

Originally, Russell used his argument to show that the set theory is not consistent.(See Malitz Section 1.11)

The axiom of regularity also rules out the set of all sets.

U ∈ U. See above.

A is not a set but is a “class”. (Von Neumann)

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

U ∈ U. See above.

Russell’s paradox

Classes

A class is an object defined by ∈, {}.

Axioms: extension and specification only.

A class is a set if it is a member of another class.

Russell’s paradox

Classes

A class is an object defined by ∈, {}.Axioms: extension and specification only.

Russell’s paradox

Classes

A class is an object defined by ∈, {}.Axioms: extension and specification only.

Russell’s paradox

The axiom of choice

Cartesian product (define later)

∏i∈I Xi := {(xi), |xi ∈ Xi for each i ∈ I}.

Axiom of Choice: The Cartesian product of a non-empty family of nonempty setsis nonempty.

Russell’s paradox

The axiom of choice

Cartesian product (define later)∏i∈I Xi := {(xi), |xi ∈ Xi for each i ∈ I}.

Russell’s paradox

The axiom of choice

Cartesian product (define later)∏i∈I Xi := {(xi), |xi ∈ Xi for each i ∈ I}.

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