Zermelo's categoricity theorem

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 ${\displaystyle \mathrm {ZFC} ^{2}}$ denote Zermelo-Fraenkel set theory, but with a second-order version of the axiom of replacement formulated as follows:^[1]

${\displaystyle \forall F\forall x\exists y\forall z(z\in y\iff \exists w(w\in x\land z=F(w)))}$

, namely the second-order universal closure of the axiom schema of replacement.^[2]p. 289 Then every model of ${\displaystyle \mathrm {ZFC} ^{2}}$ is isomorphic to a set ${\displaystyle V_{\kappa }}$ in the von Neumann hierarchy, for some inaccessible cardinal ${\displaystyle \kappa }$.^[3]

Original presentation

Zermelo originally considered a version of ${\displaystyle \mathrm {ZFC} ^{2}}$ with urelements. Rather than using the modern satisfaction relation ${\displaystyle \vDash }$, he defines a "normal domain" to be a collection of sets along with the true ${\displaystyle \in }$ relation that satisfies ${\displaystyle \mathrm {ZFC} ^{2}}$.^[4]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.^{[4]pp. 5–6[3]p. 1} Uzquiano proved that when removing replacement form ${\displaystyle {\mathsf {ZFC}}^{2}}$ and considering a second-order version of Zermelo set theory with a second-order version of separation, there exist models not isomorphic to any ${\displaystyle V_{\delta }}$ for a limit ordinal ${\displaystyle \delta >\omega }$.^[5]p. 396