Generalizations of the Kunen inconsistency

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or m...

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Bibliographic Details
Published in:Annals of pure and applied logic 2012-12, Vol.163 (12), p.1872-1890
Main Authors: Hamkins, Joel David, Kirmayer, Greg, Perlmutter, Norman Lewis
Format: Article
Language:English
Subjects:
Definability
Elementary embeddings
Forcing
Kunen inconsistency
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Summary:We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one set-forcing ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.
ISSN:0168-0072
DOI:10.1016/j.apal.2012.06.001