Core model

In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set theoretic assumptions have very special properties, most notably covering properties. Intuitively, the core model is "the largest canonical inner model there is" (Ernest Schimmerling and John R. Steel) and is typically associated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase "core model below Φ" refers to the definable inner model that exhibits the special properties under the assumption that there does not exist a cardinal satisfying Φ.

The first core model was Kurt Gödel's constructible universe L. Ronald Jensen proved the Covering Lemma for L in the 1970s under the assumption of the non-existence of zero sharp, establishing that L is the "core model below zero sharp". Together with Tony Dodd, Jensen constructed the Dodd–Jensen core model ("the core model below a measurable cardinal") and proved the covering lemma for it. Larger core models include the Mitchell core model and the Steel core model below a Woodin cardinal.

Core models are constructed by transfinite recursion from small fragments of the core model called mice. An important ingredient of the construction is the comparison lemma that allows to give a wellordering of the relevant mice.