Ideas from 'Set Theory' by Kenneth Kunen [1980], by Theme Structure

[found in 'Set Theory: Introduction to Independence Proofs' by Kunen,Kenneth [North-Holland 1980,0-444-85-401-0]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I

13030

Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II

13032

Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III

13033

Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V

13037

Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI

13038

Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII

13034

Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII

13039

Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y)))

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

13036

Choice: ∀A ∃R (R well-orders A)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence

13029

Set Existence: ∃x (x = x)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension

13031

Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L

13040

Constructibility: V = L (all sets are constructible)