A - Axioms of set theory

Summary

Axiom 0(Set Existence) There exists a set:

Axiom 1(Extensionality) If x and y have the same elements, then x is equal to y:

Axiom 2 (Comprehension scheme or schema of separation) For every formula ϕ(s,t) with free variables s and t, for every x, and for every parameter p there exists a set y = {u ∈ x: ϕ(u,p)} that contains all those u ∈ x that have property ϕ:

Axiom 3(Pairing) For any a and b there exists a set x that contains a and b:

Axiom 4(Union) For every family ℱ there exists a set U containing the union ∪ℱ of all elements of ℱ

Axiom 5(Power set) For every set X there exists a set P containing the set P(X) (the power set) of all subsets of X:

To make the statement of the next axiom more readable we introduce the following abbreviation. We say that y is a successor of x and write y = S(x) if S(x) = x ∪ {x}, that is,

Axiom 6(Infinity) (Zermelo 1908) There exists an infinite set (of some special form):

Axiom 7(Replacement scheme) (Fraenkel 1922; Skolem 1922) For every formula ϕ(s, t, U, w) with free variables s, t, U, and w, every set A, and every parameter p if ϕ(s,t,A,p) defines a function F on A by F(x) = y ⇔ ϕ(x,y,A,p), then there exists a set Y containing the range F[A] = {F(x): x ∈ A} of the function F: