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

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

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

Original presentation

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