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 $\mathrm^2$ 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.G. Uzquiano, "Models of Second-Order Zermelo Set Theory". Bulletin of Symbolic Logic, vol. 5, no. 3 (1999), pp.289--302.^p. 289 Then every model of $\mathrm^2$ is isomorphic to a set $V_\kappa$ in the von Neumann hierarchy, for some inaccessible cardinal $\kappa$., Theorem 1.

Original presentation

Zermelo originally considered a version of $\mathrm^2$ 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 $\mathrm^2$.^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 $\mathsf^2$ 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 $\delta>\omega$.A. Kanamori, "Introductory note to 1930a". In
Ernst Zermelo - Collected Works/Gesammelte Werke
' (2009), DOI 10.1007/978-3-540-79384-7.^p. 396

References

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Set theory
Theorems in the foundations of mathematics
Model theory