In mathematical set theory, 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.p. 289 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 strongly 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}} .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 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 } .p. 396
References