Zermelo's categoricity theorem

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.

Let $\mathrm {ZFC} ^{2}$ 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 $\mathrm {ZFC} ^{2}$ is isomorphic to a set $V_{\kappa }$ in the von Neumann hierarchy, for some strongly inaccessible cardinal $\kappa$ .^[3]

Original presentation

Zermelo originally considered a version of $\mathrm {ZFC} ^{2}$ 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 $\mathrm {ZFC} ^{2}$ .^[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.^[4]^pp. 5–6^[3]^p. 1 Uzquiano proved that when removing replacement from ${\mathsf {ZFC}}^{2}$ and considering a second-order version of Zermelo set theory with a second-order version of separation, there exist models not isomorphic to any $V_{\delta }$ for a limit ordinal $\delta >\omega$ .^[5]^p. 396

Zermelo's categoricity theorem

Original presentation

Related results

References

Related Articles