The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following, exists stands for exists, for all means for all, element stands for "is an element of, " ∅ for the empty set, ⇒ for implies, ⋀ for AND, ⋁ for OR, and congruent for "is equivalent to." 1. Axiom of Extensionality: If X and Y have the same elements, then X = Y. for all u(u element X congruent u element Y)⇒X = Y. 2. Axiom of the Unordered Pair: For any a and b there exists a set {a, b} that contains exactly a and b. (also called Axiom of Pairing) for all a for all b exists c for all x(x element c congruent (x = a⋁x = b)).

axiom of choice | axiom of extensionality | axiom of foundation | axiom of infinity | axiom of replacement | axiom of subsets | axiom of the power set | axiom of the unordered pair | von Neumann-Bernays-Gödel set theory | Zermelo-Fraenkel set theory | Zermelo set theory