Inner model

Summary

An inner model is a transitive class of sets that contains all ordinals and satisfies the axioms of ZF. It is a smaller universe of sets that lives inside the universe $V$ and has the same ordinals. Inner models give consistency proofs: a statement true in an inner model is consistent with ZF. The constructible universe $L$ is the smallest inner model, and the core model is a larger inner model built to hold large cardinals.