In mathematical set theory, Zermelo's categoricity theorem was proved 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 denote Zermelo–Fraenkel set theory, but with a second-order version of the axiom of replacement formulated as follows:S. Shapiro, Foundations Without Foundationalism: A Case for Second-order Logic (1991).
, 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 is isomorphic to a set in the von Neumann hierarchy, for some strongly inaccessible cardinal ., Theorem 1.
Original presentation
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 .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 from 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 .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