Condensable models of set theory

2019/10/09 by Ali Enayat, Enayat, Ali
Mathematics · #03C62 #03E30 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03C62 #msc:03E30

paper · pdf · doi:10.48550/arxiv.1910.04029

15 pages (this is revised version of a previously posted draft)

arxiv created 2021/06/18 · arxiv updated 2021/06/21

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.

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