Editing Zermelo–Fraenkel set theory (section)

== History ==
{{Main|History of set theory}}

The modern study of [[set theory]] was initiated by [[Georg Cantor]] and [[Richard Dedekind]] in the 1870s. However, the discovery of [[Paradoxes of set theory|paradoxes]] in [[naive set theory]], such as [[Russell's paradox]], led to the desire for a more rigorous form of set theory that was free of these paradoxes.

In 1908, [[Ernst Zermelo]] proposed the first [[axiomatic set theory]], [[Zermelo set theory]]. However, as first pointed out by [[Abraham Fraenkel]] in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and [[cardinal number]]s whose existence was taken for granted by most set theorists of the time, notably the cardinal number [[Aleph number#Aleph-omega|aleph-omega (<math>\aleph_{\omega}</math>)]] and the set <math>\{Z_{0},\mathcal{P}(Z_{0}),\mathcal{P}( \mathcal{P}(Z_{0}) ),\mathcal{P}( \mathcal{P}( \mathcal{P}(Z_{0}) ) ),...\},</math> where <math>Z_{0}</math> is any infinite set and <math>\mathcal{P}</math> is the [[power set]] operation.{{sfn|Ebbinghaus|2007|p=136}} Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and [[Thoralf Skolem]] independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a [[first-order logic]] whose [[atomic formula]]s were limited to set membership and identity. They also independently proposed replacing the [[axiom schema of specification]] with the [[axiom schema of replacement]]. Appending this schema, as well as the [[axiom of regularity]] (first proposed by [[John von Neumann]]),{{sfn|Halbeisen|2011|pp=62–63}} to Zermelo set theory yields the theory denoted by ''ZF''. Adding to ZF either the [[axiom of choice]] (AC) or a statement that is equivalent to it yields ZFC.