Every countable model of set theory embeds into its own constructible universe

Journal of Mathematical Logic 13 (2):1350006 (2013)
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Abstract

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations.

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Joel David Hamkins
Oxford University

References found in this work

Elementary embeddings and infinitary combinatorics.Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (3):407-413.
Non-Well-Founded Sets.Peter Aczel - 1988 - Palo Alto, CA, USA: Csli Lecture Notes.
On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
Countable models of set theories.Harvey Friedman - 1973 - In A. R. D. Mathias & Hartley Rogers, Cambridge Summer School in Mathematical Logic. New York,: Springer Verlag. pp. 539--573.
Generalizations of the Kunen inconsistency.Joel David Hamkins, Greg Kirmayer & Norman Lewis Perlmutter - 2012 - Annals of Pure and Applied Logic 163 (12):1872-1890.

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