Zermelo's categoricity theorem explained

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

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

ZFC ^{2} is isomorphic to a set V _{\kappa} in the von Neumann hierarchy, for some inaccessible cardinal \kappa .^[3]

Original presentation

Zermelo originally considered a version of

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 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.^{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 .^{[5] p. 396}

Notes and 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. 2009.07164 . Joel David Hamkins . Hans Robin Solberg . Categorical large cardinals and the tension between categoricity and set-theoretic reflection . 2020 . math.LO ., Theorem 1.
4. 2204.13754 . Maddy . Penelope . Väänänen . Jouko . Philosophical Uses of Categoricity Arguments . 2022 . 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.