Supercompact cardinal

Summary

A supercompact cardinal is a cardinal $\kappa$ that is the critical point of many elementary embeddings $j \colon V \to M$. For each length, one of them has a target $M$ closed under sequences of that length. It strengthens the embedding form of measurability, which asks only for closure under sequences of length $\kappa$. Supercompact cardinals are Woodin cardinals and lie high in the hierarchy of large cardinals. They are used to prove the consistency of strong axioms such as the proper forcing axiom.