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In [[set theory]], the '''core model''' is a definable [[inner model]] of the [[von Neumann universe|universe]] of all [[Set (mathematics)|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 lemma|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.
==History==
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]].
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