# 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 and there is a set whose only elements are and . (Note that taking , this gives you the possibility of forming the singleton set .)

**Axiom 4:** For any set there's a set whose elements are the elements of the elements of A. (If then this set is what's called the *union* of and , whose elements are all the elements of together with the elements of . For example if and then .)

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

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

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

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