Zermelo's categoricity theorem

In mathematical set theory, Zermelo's categoricity theorem was proven by Ernst Zermelo in 1930. It states that all models of a certain second-order version of the Zermelo–Fraenkel axioms of set theory are isomorphic to a member of a certain class of sets.

Statement

Let denote Zermelo–Fraenkel set theory, but with a second-order version of the axiom of replacement formulated as follows:
, namely the second-order universal closure of the axiom schema of replacement.^{p. 289} Then every model of is isomorphic to a set in the von Neumann hierarchy, for some strongly inaccessible cardinal.

Original presentation

Zermelo originally considered a version of with urelements. Rather than using the modern satisfaction relation, he defines a "normal domain" to be a collection of sets along with the true relation that satisfies.^{p. 9}

Related results

Dedekind proved that the second-order Peano axioms hold in a model if and only if the model is isomorphic to the true natural numbers.^{pp. 5–6}^{p. 1} Uzquiano proved that when removing replacement from and considering a second-order version of Zermelo set theory with a second-order version of separation, there exist models not isomorphic to any for a limit ordinal.^{p. 396}