An extension of a theorem of Zermelo

TL;DR

This paper extends Zermelo's 1930 theorem by showing that if a model satisfies the Zermelo-Fraenkel axioms with two different membership relations, then the structures are isomorphic and this isomorphism is definable within the combined structure.

Contribution

It generalizes Zermelo's theorem to first-order models with two membership relations, establishing their isomorphism and definability within the combined structure.

Findings

Models with two membership relations satisfying ZF axioms are isomorphic.

The isomorphism between such models is definable within the combined structure.

Extends classical Zermelo's theorem to a broader first-order context.

Abstract

We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E'). This extends Zermelo's 1930 theorem about second order ZFC.