Condensable models of set theory

TL;DR

This paper investigates models of set theory called condensable models, which can be embedded into their own initial segments, revealing new structural properties and characterizations under certain hypotheses.

Contribution

It introduces the concept of condensable models, proves their existence under specific conditions, and provides characterizations of countable condensable models of ZF.

Findings

Existence of condensable models of ZFC with all definable elements in the well-founded part.

Characterizations of countable condensable models of ZF.

Conditions under which condensable models can be constructed.

Abstract

We study models M of set theory that are "condensable", in the sense that there is an "ordinal" v of M such that the rank initial segment of M determined by v is both isomorphic to M, and also an elementary submodel of M for infinitary formulae in the well-founded part of M. We prove, assuming a modest set theoretic hypothesis, that there are condensable models M of ZFC such that every definable element of M is in the well-founded part of M. We also provide various characterizations of countable condensable models of ZF.