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 I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.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.