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{{distinguish|Core product}}
{{More sources needed|date=September 2024}}
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",<ref>{{cite arXiv|eprint=math/9702206v1 |last1=Schimmerling |first1=Ernest |last2=Steel |first2=John R. |title=The maximality of the core model |date=1997 }}</ref> (here "canonical" is an undefined term)<ref>G. Sargsyan, "[https://www.math.uni-bonn.de/ag/logik/events/young-set-theory-2011/Slides/Grigor_Sargsyan_slides.pdf An invitation to inner model theory]". Talk slides, Young Set Theory Meeting, 2011.</ref><sup>p. 28</sup> 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==
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Partial results for the conjecture are that:
#If there is no inner model with a Woodin cardinal, then K exists.
#If (boldface) Σ<sup>1</sup><sub>n</sub> [[determinacy]] (n is finite) holds in every generic extension of V, but there is no iterable inner model with n Woodin cardinals, then K exists.
#If there is a measurable cardinal κ, then either K<sup>c</sup> below κ exists, or there is an ω<sub>1</sub>+1 iterable model with measurable limit λ of both Woodin cardinals and cardinals strong up to λ.
If V has Woodin cardinals but not cardinals strong past a Woodin one, then under appropriate circumstances (a candidate for) K can be constructed by constructing K below each Woodin cardinal (and below the class of all ordinals) κ but above that K as constructed below the [[Infimum and supremum|supremum]] of Woodin cardinals below κ. The candidate core model is not fully iterable (iterability fails at Woodin cardinals) or generically absolute, but otherwise behaves like K.
==References==
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