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 core model program seeks to analyze large cardinal axioms by determining the core models below them.

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". The work of Solovay isolated another core model L[U], for U an ultrafilter on a measurable cardinal (and its associated "sharp", zero dagger). 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 and a generalized covering lemma for L[U].

Mitchell used coherent sequences of measures to develop core models containing multiple or higher-order measurables. Still later, the Steel core model used extenders and iteration trees to construct a core model below a Woodin cardinal.

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