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.