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

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Let   denote Zermelo-Fraenkel set theory, but with a second-order version of the axiom of replacement formulated as follows:[1]

 

, namely the second-order universal closure of the axiom schema of replacement.[2]p. 289 Then every model of   is isomorphic to a set   in the von Neumann hierarchy, for some inaccessible cardinal  .[3]

Original presentation

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Zermelo originally considered a version of   with urelements. Rather than using the modern satisfaction relation  , he defines a "normal domain" to be a collection of sets along with the true   relation that satisfies  .[4]p. 9

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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   and considering a second-order version of Zermelo set theory with a second-order version of separation, there exist models not isomorphic to any   for a limit ordinal  .[5]p. 396

References

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  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. ^ a b 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. ^ a b 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.