• Corpus ID: 237347066

Mutually embeddable models of ZFC

@inproceedings{Eskew2021MutuallyEM,
  title={Mutually embeddable models of ZFC},
  author={Monroe Eskew and Sy-David Friedman and Yair Hayut and Farmer Schlutzenberg},
  year={2021}
}

• Monroe Eskew, S. Friedman, +1 author Farmer Schlutzenberg
• Published 27 August 2021
• Mathematics

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems. One of Ken Kunen’s best-known and most striking results is that there is no elementary embedding of the universe of sets into itself other than the identity. Kunen’s result is best understood in a theory that includes proper classes as genuine objects, such as von Neumann-Gödel-Bernays set theory (NBG), which we take in this article as our background theory. The argument proceeds by assuming there is a nontrivial j : V → V and using Replacement and Comprehension for formulas involving the parameter j, arriving at a contradiction. It does not exclude the possibility of the existence of an inner model M of Zermelo-Fraenkel set theory (ZFC) and a nontrivial elementary j : M → M , where M is not the class of sets of an NBG-universe that includes the class j. Indeed, Kunen’s early work on 0 showed an equivalence between the nontrivial self-embeddability of L and the existence of a certain real number external to L. In this paper we take up a related theme, that of mutual embeddability between models of set theory. In [4], the first two authors asked whether there can be two transitive models M 6= N of ZFC (necessarily with the same ordinals) such that each is elementarily embeddable into the other; that is, such that there are elementary embeddings j : M → N and k : N → M . We say such models are mutually embeddable. We answer the question affirmatively with the following result: Theorem 1. The following are equivalent: • 0 exists. • There are transitive models M 6= N of ZFC containing all ordinals which are mutually embeddable. There are natural variants of the original question, given by making more demands on either the models M,N or the embeddings j, k witnessing mutual embeddability. For example, what can hold regarding whether M ⊆ N or N ⊆ M? To what extent can j, k be amenable to M,N (that is, how much of j, k can belong to M,N)? Can we arrange that j ↾ Ord = k ↾ Ord? Can we arrange that The first two authors would like to acknowledge the support of the Austrian Science Fund (FWF) through Project P28420. The third author’s research was partially supported by the FWF Lise Meitner Grant (2650N35). The fourth author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.