Generalizations of the Kunen Inconsistency

TL;DR

This paper extends the Kunen inconsistency to various models and embeddings, showing that no nontrivial elementary embedding exists between certain models of set theory, including forcing extensions and definable classes.

Contribution

It provides a unified framework for multiple generalizations of the Kunen inconsistency, including new results and consolidating known folklore and unpublished findings.

Findings

No elementary embedding from V to V[G] or vice versa.

No elementary embedding between different ground models or between V and HOD.

Results hold even without the axiom of choice and include generic and definable embeddings.

Abstract

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 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 We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.