The ZF axioms

Zermelo and Fraenkel expressed their axioms in the formal language of set theory, but we can translate them into ordinary language.

Axiom 1: Two sets are equal if and only if they have the same elements.

Axiom 2: There is a set with no elements called the empty set.

Axiom 3: For any sets A and B there is a set whose only elements are A and B. (Note that taking A=B, this gives you the possibility of forming the singleton set {A}.)

Axiom 4: For any set A there's a set whose elements are the elements of the elements of A. (If A={B,C} then this set is what's called the union BC of B and C, whose elements are all the elements of B together with the elements of C. For example if B={1,2} and C={3,4}, then BC={1,2,3,4}.)

Axiom 5: For any set A there is a set, called the power set of A, whose elements are the subsets of A. (A subset of A is a set whose elements are all contained in A. Power sets are an important ingredient of Cantor's hierarchy of infinities.)

Axiom 6: For any function f defined on a set A, the values f(a), where a is an element of A, also form a set.

Axiom 7: Every non-empty set A has a member whose elements are all different from the elements of A. (This axiom insures that if you take an element B of A and then an element C of B, and so on, this process stops after finitely many steps.)

Axiom 8: There is an infinite set such that if A is in this set, the set A{A} is also in the set.

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