Kunen's inconsistency

Kunen's inconsistency is a theorem proved by Kenneth Kunen which proves that certain large cardinals, which were previously believed to be natural generalizations of weaker large cardinals, are inconsistent. The proof uses the axiom of choice in a crucial way - it is believed that the large cardinals the theorem rules out may still be able to exist in ZF. It originally proved: "there is no nontrivial elementary embedding \(j: V \to V\), therefore there do not exist Reinhardt cardinals". However, its proof can also be generalized to show:

• There are no \(\omega\)-huge cardinals. That is, there is no inner model \(M\) and nontrivial elementary embedding \(j: V \to M\) so that \(M^\lambda \subseteq M\), where \(\lambda = \sup\{j^n(\kappa): n < \omega\}\).
• For all limit cardinals \(\lambda\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\).

However, it is believed that it can not be generalized to show the nonexistence of a nontrivial elementary embedding \(j: V_{\lambda+1} \to V_{\lambda+1}\). Therefore, the rank-into-rank cardinals are believed to be consistent, but barely teetering on the brink of inconsistency.

Kunen's inconsistency also implies that if \(\kappa\) is supercompact, \(N\) is a weak extender model for \(\kappa\)'s supercompactness, and \(j: N \to N\) is a nontrivial elementary embedding, then the critical point of \(j\) is less than \(\kappa\). It is known that there must be such an elementary embedding.