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

$\mathrm{\forall }F\mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z(z\in y⟺\mathrm{\exists }w(w\in x\wedge z=F(w)))$

, namely the second-order universal closure of the axiom schema of replacement.^p. 289 Then every model of ${\mathrm{Z}\mathrm{F}\mathrm{C}}^2$ 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 ${\mathrm{Z}\mathrm{F}\mathrm{C}}^2$ with urelements. Rather than using the modern satisfaction relation $\models$, he defines a "normal domain" to be a collection of sets along with the true $\in$ relation that satisfies ${\mathrm{Z}\mathrm{F}\mathrm{C}}^2$.^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 ${𝖹𝖥𝖢}^2$ 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