Lecture 9: Axiomatic Set Theory

Home , Axiom of pairing, Axiom of union

December 16, 2014

Lecture 9: Axiomatic Set Theory Key points of today’s lecture:

I The iterative concept of set.

I The language of set theory (LOST).

I The axioms of Zermelo-Fraenkel set theory (ZFC).

I Justiﬁcation of the axioms based on the iterative concept of set.

Today

Lecture 9: Axiomatic Set Theory I The iterative concept of set.

I The language of set theory (LOST).

I The axioms of Zermelo-Fraenkel set theory (ZFC).

I Justiﬁcation of the axioms based on the iterative concept of set.

Today

Key points of today’s lecture:

Lecture 9: Axiomatic Set Theory I The language of set theory (LOST).

I The axioms of Zermelo-Fraenkel set theory (ZFC).

I Justiﬁcation of the axioms based on the iterative concept of set.

Today

Key points of today’s lecture:

I The iterative concept of set.

Lecture 9: Axiomatic Set Theory I The axioms of Zermelo-Fraenkel set theory (ZFC).

I Justiﬁcation of the axioms based on the iterative concept of set.

Today

Key points of today’s lecture:

I The iterative concept of set.

I The language of set theory (LOST).

Lecture 9: Axiomatic Set Theory I Justiﬁcation of the axioms based on the iterative concept of set.

Today

Key points of today’s lecture:

I The iterative concept of set.

I The language of set theory (LOST).

I The axioms of Zermelo-Fraenkel set theory (ZFC).

Lecture 9: Axiomatic Set Theory Today

Key points of today’s lecture:

I The iterative concept of set.

I The language of set theory (LOST).

I The axioms of Zermelo-Fraenkel set theory (ZFC).

I Justiﬁcation of the axioms based on the iterative concept of set.

Lecture 9: Axiomatic Set Theory The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership. Note that the atomic formulas of LOST are of the form

x = y and x ∈ y.

We will write x 6= y and x ∈/ y for the negation of x = y and x ∈ y, respectively.

The language of set theory, I

Lecture 9: Axiomatic Set Theory Note that the atomic formulas of LOST are of the form

x = y and x ∈ y.

We will write x 6= y and x ∈/ y for the negation of x = y and x ∈ y, respectively.

The language of set theory, I

The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership.

Lecture 9: Axiomatic Set Theory x = y and x ∈ y.

We will write x 6= y and x ∈/ y for the negation of x = y and x ∈ y, respectively.

The language of set theory, I

The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership. Note that the atomic formulas of LOST are of the form

Lecture 9: Axiomatic Set Theory We will write x 6= y and x ∈/ y for the negation of x = y and x ∈ y, respectively.

The language of set theory, I

The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership. Note that the atomic formulas of LOST are of the form

x = y and x ∈ y.

Lecture 9: Axiomatic Set Theory x 6= y and x ∈/ y for the negation of x = y and x ∈ y, respectively.

The language of set theory, I

The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership. Note that the atomic formulas of LOST are of the form

x = y and x ∈ y.

We will write

Lecture 9: Axiomatic Set Theory for the negation of x = y and x ∈ y, respectively.

The language of set theory, I

The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership. Note that the atomic formulas of LOST are of the form

x = y and x ∈ y.

We will write x 6= y and x ∈/ y

Lecture 9: Axiomatic Set Theory The language of set theory, I

The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, ∈, called membership. Note that the atomic formulas of LOST are of the form

x = y and x ∈ y.

We will write x 6= y and x ∈/ y for the negation of x = y and x ∈ y, respectively.

Lecture 9: Axiomatic Set Theory In addition to the variables of our formal language, v1, v2,..., we will also use the letters

x, y, z, u, w, x1, x2,..., y1, y2,..., w1, w2,...

to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: ∨, ∧, ↔ and ∃ for “or”, “and”, “if and only if”, and “there exists”.

The language of set theory, II

Lecture 9: Axiomatic Set Theory x, y, z, u, w, x1, x2,..., y1, y2,..., w1, w2,...

to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: ∨, ∧, ↔ and ∃ for “or”, “and”, “if and only if”, and “there exists”.

The language of set theory, II

In addition to the variables of our formal language, v1, v2,..., we will also use the letters

Lecture 9: Axiomatic Set Theory to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: ∨, ∧, ↔ and ∃ for “or”, “and”, “if and only if”, and “there exists”.

The language of set theory, II

In addition to the variables of our formal language, v1, v2,..., we will also use the letters

x, y, z, u, w, x1, x2,..., y1, y2,..., w1, w2,...

Lecture 9: Axiomatic Set Theory We will allow ourselves to use the standard abbreviations: ∨, ∧, ↔ and ∃ for “or”, “and”, “if and only if”, and “there exists”.

The language of set theory, II

In addition to the variables of our formal language, v1, v2,..., we will also use the letters

x, y, z, u, w, x1, x2,..., y1, y2,..., w1, w2,...

to stand for arbitrary formal variables.

Lecture 9: Axiomatic Set Theory The language of set theory, II

In addition to the variables of our formal language, v1, v2,..., we will also use the letters

x, y, z, u, w, x1, x2,..., y1, y2,..., w1, w2,...

to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: ∨, ∧, ↔ and ∃ for “or”, “and”, “if and only if”, and “there exists”.

Lecture 9: Axiomatic Set Theory 0. Axiom of Set Existence:

∃v1v1 = v1.

(There is a set.) 1. Axiom of Extensionality:

∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

(Sets that have the same members are identical.)

Axioms 0 and 1

Lecture 9: Axiomatic Set Theory ∃v1v1 = v1.

(There is a set.) 1. Axiom of Extensionality:

∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

(Sets that have the same members are identical.)

Axioms 0 and 1

0. Axiom of Set Existence:

Lecture 9: Axiomatic Set Theory (There is a set.) 1. Axiom of Extensionality:

∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

(Sets that have the same members are identical.)

Axioms 0 and 1

0. Axiom of Set Existence:

∃v1v1 = v1.

Lecture 9: Axiomatic Set Theory 1. Axiom of Extensionality:

∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

(Sets that have the same members are identical.)

Axioms 0 and 1

0. Axiom of Set Existence:

∃v1v1 = v1.

(There is a set.)

Lecture 9: Axiomatic Set Theory ∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

(Sets that have the same members are identical.)

Axioms 0 and 1

0. Axiom of Set Existence:

∃v1v1 = v1.

(There is a set.) 1. Axiom of Extensionality:

Lecture 9: Axiomatic Set Theory (Sets that have the same members are identical.)

Axioms 0 and 1

0. Axiom of Set Existence:

∃v1v1 = v1.

(There is a set.) 1. Axiom of Extensionality:

∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

Lecture 9: Axiomatic Set Theory Axioms 0 and 1

0. Axiom of Set Existence:

∃v1v1 = v1.

(There is a set.) 1. Axiom of Extensionality:

∀v1∀v2(∀v3(v3 ∈ v1 ↔ v3 ∈ v2) → v1 = v2).

(Sets that have the same members are identical.)

Lecture 9: Axiomatic Set Theory 2. Axiom of Foundation:

∀v1(∃v2 v2 ∈ v1 → ∃v2(v2 ∈ v1 ∧ ∀v3 v3 ∈/ v1 ∨ v3 ∈/ v2)).

(Every non-empty set has a member which has no members in common with it.)

Axiom 2

Lecture 9: Axiomatic Set Theory ∀v1(∃v2 v2 ∈ v1 → ∃v2(v2 ∈ v1 ∧ ∀v3 v3 ∈/ v1 ∨ v3 ∈/ v2)).

(Every non-empty set has a member which has no members in common with it.)

Axiom 2

2. Axiom of Foundation:

Lecture 9: Axiomatic Set Theory (Every non-empty set has a member which has no members in common with it.)

Axiom 2

2. Axiom of Foundation:

∀v1(∃v2 v2 ∈ v1 → ∃v2(v2 ∈ v1 ∧ ∀v3 v3 ∈/ v1 ∨ v3 ∈/ v2)).

Lecture 9: Axiomatic Set Theory Axiom 2

2. Axiom of Foundation:

∀v1(∃v2 v2 ∈ v1 → ∃v2(v2 ∈ v1 ∧ ∀v3 v3 ∈/ v1 ∨ v3 ∈/ v2)).

(Every non-empty set has a member which has no members in common with it.)

Lecture 9: Axiomatic Set Theory 3. Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

(For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) !

Axiom 3

Lecture 9: Axiomatic Set Theory ∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

(For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) !

Axiom 3

3. Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

Lecture 9: Axiomatic Set Theory (For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) !

Axiom 3

3. Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

Lecture 9: Axiomatic Set Theory !

Axiom 3

3. Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

(For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.)

Lecture 9: Axiomatic Set Theory Axiom 3

3. Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

(For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) !

Lecture 9: Axiomatic Set Theory 4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Axioms 4 and 5

Lecture 9: Axiomatic Set Theory ∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Axioms 4 and 5

4. Axiom of Pairing:

Lecture 9: Axiomatic Set Theory (For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Axioms 4 and 5

4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

Lecture 9: Axiomatic Set Theory 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Axioms 4 and 5

4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.)

Lecture 9: Axiomatic Set Theory ∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Axioms 4 and 5

4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

Lecture 9: Axiomatic Set Theory (For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Axioms 4 and 5

4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

Lecture 9: Axiomatic Set Theory !

Axioms 4 and 5

4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.)

Lecture 9: Axiomatic Set Theory Axioms 4 and 5

4. Axiom of Pairing:

∀v1∀v2∃v3(v1 ∈ v3 ∧ v2 ∈ v3).

(For any two sets, there is a set to which they both belong, i.e., of which they are both members.) 5. Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

(For any set, there is another set to which all members of members of the ﬁrst set belongs.) !

Lecture 9: Axiomatic Set Theory In the statement of the next axiom, the notation ∃!vi is shorthand for the obvious way of expressing “there is exactly one vi ”. 6. Axiom Schema of Replacement: For each LOST formula ϕ(x, y, z, w1,..., wn) with free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z(∀x(x ∈ z → ∃!yϕ) → ∃u∀x(x ∈ z → ∃y(y ∈ u ∧ ϕ))).

(For any set z and relation R (which must be expressible by some ﬁrst order formula ϕ of LOST), if each member x of z bears the relation R to exactly one set yx , then there is a set to which all these yx belong.)

Axiom 6

Lecture 9: Axiomatic Set Theory 6. Axiom Schema of Replacement: For each LOST formula ϕ(x, y, z, w1,..., wn) with free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z(∀x(x ∈ z → ∃!yϕ) → ∃u∀x(x ∈ z → ∃y(y ∈ u ∧ ϕ))).

Axiom 6

In the statement of the next axiom, the notation ∃!vi is shorthand for the obvious way of expressing “there is exactly one vi ”.

Lecture 9: Axiomatic Set Theory For each LOST formula ϕ(x, y, z, w1,..., wn) with free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z(∀x(x ∈ z → ∃!yϕ) → ∃u∀x(x ∈ z → ∃y(y ∈ u ∧ ϕ))).

Axiom 6

In the statement of the next axiom, the notation ∃!vi is shorthand for the obvious way of expressing “there is exactly one vi ”. 6. Axiom Schema of Replacement:

Lecture 9: Axiomatic Set Theory ∀w1 · · · ∀wn∀z(∀x(x ∈ z → ∃!yϕ) → ∃u∀x(x ∈ z → ∃y(y ∈ u ∧ ϕ))).

Axiom 6

In the statement of the next axiom, the notation ∃!vi is shorthand for the obvious way of expressing “there is exactly one vi ”. 6. Axiom Schema of Replacement: For each LOST formula ϕ(x, y, z, w1,..., wn) with free variables among those shown, the following is an axiom:

Lecture 9: Axiomatic Set Theory (For any set z and relation R (which must be expressible by some ﬁrst order formula ϕ of LOST), if each member x of z bears the relation R to exactly one set yx , then there is a set to which all these yx belong.)

Axiom 6

∀w1 · · · ∀wn∀z(∀x(x ∈ z → ∃!yϕ) → ∃u∀x(x ∈ z → ∃y(y ∈ u ∧ ϕ))).

Lecture 9: Axiomatic Set Theory Axiom 6

∀w1 · · · ∀wn∀z(∀x(x ∈ z → ∃!yϕ) → ∃u∀x(x ∈ z → ∃y(y ∈ u ∧ ϕ))).

Lecture 9: Axiomatic Set Theory For the next axiom, let S(x) = x ∪ {x}. 7. Axiom of Inﬁnity:

∃v1(∅ ∈ v0 ∧ ∀v1(v1 ∈ v0 → S(v1) ∈ v0)).

(There is a set that has the empty set as a member and is closed under the operation S.)

Axiom 7

Lecture 9: Axiomatic Set Theory 7. Axiom of Inﬁnity:

∃v1(∅ ∈ v0 ∧ ∀v1(v1 ∈ v0 → S(v1) ∈ v0)).

(There is a set that has the empty set as a member and is closed under the operation S.)

Axiom 7

For the next axiom, let S(x) = x ∪ {x}.

Lecture 9: Axiomatic Set Theory ∃v1(∅ ∈ v0 ∧ ∀v1(v1 ∈ v0 → S(v1) ∈ v0)).

(There is a set that has the empty set as a member and is closed under the operation S.)

Axiom 7

For the next axiom, let S(x) = x ∪ {x}. 7. Axiom of Inﬁnity:

Lecture 9: Axiomatic Set Theory (There is a set that has the empty set as a member and is closed under the operation S.)

Axiom 7

For the next axiom, let S(x) = x ∪ {x}. 7. Axiom of Inﬁnity:

∃v1(∅ ∈ v0 ∧ ∀v1(v1 ∈ v0 → S(v1) ∈ v0)).

Lecture 9: Axiomatic Set Theory Axiom 7

For the next axiom, let S(x) = x ∪ {x}. 7. Axiom of Inﬁnity:

∃v1(∅ ∈ v0 ∧ ∀v1(v1 ∈ v0 → S(v1) ∈ v0)).

(There is a set that has the empty set as a member and is closed under the operation S.)

Lecture 9: Axiomatic Set Theory For the next axiom, let “v3 ⊆ v1” abbreviate “∀v4(v4 ∈ v3 → v4 ∈ v1)”. 8. Axiom of Power Set:

∀v1∃v2∀v3(v3 ⊆ v1 → v3 ∈ v2).

(For any set, there is a set to which all subsets of that set belong.)

Axiom 8

Lecture 9: Axiomatic Set Theory 8. Axiom of Power Set:

∀v1∃v2∀v3(v3 ⊆ v1 → v3 ∈ v2).

(For any set, there is a set to which all subsets of that set belong.)

Axiom 8

For the next axiom, let “v3 ⊆ v1” abbreviate “∀v4(v4 ∈ v3 → v4 ∈ v1)”.

Lecture 9: Axiomatic Set Theory ∀v1∃v2∀v3(v3 ⊆ v1 → v3 ∈ v2).

(For any set, there is a set to which all subsets of that set belong.)

Axiom 8

For the next axiom, let “v3 ⊆ v1” abbreviate “∀v4(v4 ∈ v3 → v4 ∈ v1)”. 8. Axiom of Power Set:

Lecture 9: Axiomatic Set Theory (For any set, there is a set to which all subsets of that set belong.)

Axiom 8

For the next axiom, let “v3 ⊆ v1” abbreviate “∀v4(v4 ∈ v3 → v4 ∈ v1)”. 8. Axiom of Power Set:

∀v1∃v2∀v3(v3 ⊆ v1 → v3 ∈ v2).

Lecture 9: Axiomatic Set Theory Axiom 8

For the next axiom, let “v3 ⊆ v1” abbreviate “∀v4(v4 ∈ v3 → v4 ∈ v1)”. 8. Axiom of Power Set:

∀v1∃v2∀v3(v3 ⊆ v1 → v3 ∈ v2).

(For any set, there is a set to which all subsets of that set belong.)

Lecture 9: Axiomatic Set Theory 9. Axiom of Choice:

∀v1(∀v2∀v3((v2 ∈ v1∧v3 ∈ v1)

→ (v2 6= ∅ ∧ (v2 = v3 ∨ v2 ∩ v3 = ∅)))

→ ∃v4∀v5(v5 ∈ v1 → ∃!v6v6 ∈ v4 ∩ v5))

(If x is a set of pairwise disjoint non-empty sets, then there is a set that has exactly one member in common with each member of x.) !!

Axiom 9

Lecture 9: Axiomatic Set Theory ∀v1(∀v2∀v3((v2 ∈ v1∧v3 ∈ v1)

→ (v2 6= ∅ ∧ (v2 = v3 ∨ v2 ∩ v3 = ∅)))

→ ∃v4∀v5(v5 ∈ v1 → ∃!v6v6 ∈ v4 ∩ v5))

(If x is a set of pairwise disjoint non-empty sets, then there is a set that has exactly one member in common with each member of x.) !!

Axiom 9

9. Axiom of Choice:

Lecture 9: Axiomatic Set Theory (If x is a set of pairwise disjoint non-empty sets, then there is a set that has exactly one member in common with each member of x.) !!

Axiom 9

9. Axiom of Choice:

∀v1(∀v2∀v3((v2 ∈ v1∧v3 ∈ v1)

→ (v2 6= ∅ ∧ (v2 = v3 ∨ v2 ∩ v3 = ∅)))

→ ∃v4∀v5(v5 ∈ v1 → ∃!v6v6 ∈ v4 ∩ v5))

Lecture 9: Axiomatic Set Theory !!

Axiom 9

9. Axiom of Choice:

∀v1(∀v2∀v3((v2 ∈ v1∧v3 ∈ v1)

→ (v2 6= ∅ ∧ (v2 = v3 ∨ v2 ∩ v3 = ∅)))

→ ∃v4∀v5(v5 ∈ v1 → ∃!v6v6 ∈ v4 ∩ v5))

(If x is a set of pairwise disjoint non-empty sets, then there is a set that has exactly one member in common with each member of x.)

Lecture 9: Axiomatic Set Theory Axiom 9

9. Axiom of Choice:

∀v1(∀v2∀v3((v2 ∈ v1∧v3 ∈ v1)

→ (v2 6= ∅ ∧ (v2 = v3 ∨ v2 ∩ v3 = ∅)))

→ ∃v4∀v5(v5 ∈ v1 → ∃!v6v6 ∈ v4 ∩ v5))

(If x is a set of pairwise disjoint non-empty sets, then there is a set that has exactly one member in common with each member of x.) !!

Lecture 9: Axiomatic Set Theory GROUP 1: Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

GROUP 2: Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

GROUP 3: Axiom of Inﬁnity:

∃v1(∅ ∈ v1 ∧ ∀v2(v2 ∈ v1 → S(v2) ∈ v1)).

Work in groups to justify

Lecture 9: Axiomatic Set Theory !!

Work in groups to justify

GROUP 1: Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

GROUP 2: Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

GROUP 3: Axiom of Inﬁnity:

∃v1(∅ ∈ v1 ∧ ∀v2(v2 ∈ v1 → S(v2) ∈ v1)).

Lecture 9: Axiomatic Set Theory Work in groups to justify

GROUP 1: Axiom Schema of Comprehension: For each LOST formula ϕ(x, z, w1,..., wn) with the (distinct) free variables among those shown, the following is an axiom:

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

GROUP 2: Axiom of Union:

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

GROUP 3: Axiom of Inﬁnity:

∃v1(∅ ∈ v1 ∧ ∀v2(v2 ∈ v1 → S(v2) ∈ v1)).

Lecture 9: Axiomatic Set Theory Group 1: Justiﬁcation of Comprehension

∀w1 · · · ∀wn∀z∃y∀x(x ∈ y ↔ (x ∈ z ∧ ϕ)).

Lecture 9: Axiomatic Set Theory Group 2: Justiﬁcation of Union

∀v1∃v2∀v3∀v4((v4 ∈ v3 ∧ v3 ∈ v1) → v4 ∈ v2).

Lecture 9: Axiomatic Set Theory Group 3: Justiﬁcation of Inﬁnity

∃v1(∅ ∈ v1 ∧ ∀v2(v2 ∈ v1 → S(v2) ∈ v1)).

Lecture 9: Axiomatic Set Theory