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}

References

1. S. Shapiro, Foundations Without Foundationalism: A Case for Second-order Logic (1991).
2. G. Uzquiano, "Models of Second-Order Zermelo Set Theory". Bulletin of Symbolic Logic, vol. 5, no. 3 (1999), pp.289--302.
3. Joel David Hamkins; Hans Robin Solberg (2020). "Categorical large cardinals and the tension between categoricity and set-theoretic reflection". arXiv: 2009.07164 [math.LO]., Theorem 1.
4. Maddy, Penelope; Väänänen, Jouko (2022). "Philosophical Uses of Categoricity Arguments". arXiv: 2204.13754 [math.LO].
5. A. Kanamori, "Introductory note to 1930a". In Ernst Zermelo - Collected Works/Gesammelte Werke (2009), DOI 10.1007/978-3-540-79384-7.