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Zermelo's categoricity theorem

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

∀ F ∀ x ∃ y ∀ z ( z ∈ y ⟺ ∃ w ( w ∈ x ∧ z = F ( w ) ) ) {\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. Then every model of Z F C 2 {\displaystyle \mathrm {ZFC} ^{2}} is isomorphic to a set V κ {\displaystyle V_{\kappa }} in the von Neumann hierarchy, for some inaccessible cardinal κ {\displaystyle \kappa } .

Original presentation

Zermelo originally considered a version of Z F C 2 {\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 Z F C 2 {\displaystyle \mathrm {ZFC} ^{2}} .

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. Uzquiano proved that when removing replacement form Z F C 2 {\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 V δ {\displaystyle V_{\delta }} for a limit ordinal δ > ω {\displaystyle \delta >\omega } .

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 ., Theorem 1.
  4. ^ Maddy, Penelope; Väänänen, Jouko (2022). "Philosophical Uses of Categoricity Arguments". arXiv:2204.13754 .
  5. A. Kanamori, "Introductory note to 1930a". In Ernst Zermelo - Collected Works/Gesammelte Werke (2009), DOI 10.1007/978-3-540-79384-7.
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