The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers, exists x(∅ element x⋀ for all y element x(y' element x)), where exists denotes exists, ∅ is the empty set, ⋀ is logical AND, for all means for all, and element denotes "is an element of". Following von Neumann, 0 = ∅, 1 = 0' = {0}, 2 = 1' = {0, 1}, 3 = 2' = {0, 1, 2}, ....

Zermelo-Fraenkel set theory