ZFC refutation of cardinal-preserving embeddings from V into N

Determine whether ZFC proves that there is no nontrivial cardinal preserving elementary embedding j: V → N between transitive models of ZFC; that is, ascertain if ZFC alone refutes the existence of any elementary embedding from the universe V into a transitive model N such that Card^V = Card^N.

Background

The paper proves that no nontrivial cardinal-preserving elementary embedding j: M → V exists, thereby answering Caicedo’s question in the case where the target model is the universe V. Prior results showed that PFA rules out the case N = V and that certain large-cardinal hypotheses refute the case M = V, but a general ZFC refutation for embeddings from V to some N had not been established.

The authors note that while the existence of a cardinal-preserving embedding j: V → N has significant consistency strength (implying the consistency of ZFC plus the existence of a strongly compact cardinal), it is not presently known whether ZFC alone rules out such embeddings. This forms a central remaining uncertainty after their main theorem.