Axioms of set theories

1. The Axiom of Extensionality

   • The Axiom of Extension

   • The Axiom of Extent

   • ∀x . (x ∈ A ⟺ x ∈ B) ⟺ A = B

   • defines sets (as objects determined by their elements)

   • defines set equality through subset (inclusion) relation

2. The Axiom of Union (flat set: containing all elements of nested sets)

3. The axiom of specification

4. The Axiom of Pairing (pair of sets as elements in third)

5. The axiom of powers

6. The axiom of infinity

7. The axiom of substitution

8. The axiom of choice (AC) (choose 1 elem out of any non-empty subset)

9. The Axiom of well-ordering (added to turn ZF into ZFC)

10. The Axiom of Regularity (disjoint sets)

11. The axiom of foundation (disjoint sets)

12. Axiom Schema of Specification

13. Axiom of Union

    for every collection of sets, there exists a set that contains all the elements that belong to at least one of the sets in the collection.

14. Axiom of Replacement

    for any set $S$, there exists a set $x$ such that, for any element $y$ of $S$, if there exists an element $z$ satisfying the condition $P(y,z)$, where $P(y,z)$ is a propositional function, then such $z$ appear in $x$.

15. Axiom of the Empty Set

    there exists a set that has no elements.

16. Axiom of Subsets

    for every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.

17. Axiom of Powerset

    for each set there exists a collection of sets that contains amongst its elements all the subsets of the given set.

18. Axiom of Infinity

    there exists a set containing a set with no elements and the successor of each of its elements.

19. Axiom of Foundation

    for all non-null sets, there is an element of the set that shares no member with the set.

20. Axiom of Choice

    for every set, we can provide a mechanism for choosing one element of any non-empty subset of the set.