Generalizations of the Kunen Inconsistency, Singapore 2011

A talk at the Prospects of Infinity: Workshop on Set Theory  at the National University of Singapore, July 18-22, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others.  For example, there is no nontrivial 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 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 from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V.  Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice.  I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.  This is joint work with Greg Kirmayer and Norman Perlmutter.

SlidesArticle 

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