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 F is a large cardinal notion, then the phrase "core model below F" refers to the definable inner model that exhibits the special properties under the assumption that there does not exist a cardinal satisfying F. The core model program seeks to analyze large cardinal axioms by determining the core models below them.