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{{distinguish|Core product}}

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.

{{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 (ErnestarXiv|eprint=math/9702206v1 |last1=Schimmerling and|first1=Ernest [[|last2=Steel |first2=John R. Steel]]|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>^{p. 28} 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". 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 [[extender (set theory)|extender]]s and iteration trees to construct a core model below a [[Woodin cardinal]].

==Construction of core models==

Core models are constructed by [[transfinite recursion]] from small fragments of the core model called [[mouse (Set Theory)|mice]]. An important ingredient of the construction is the comparison lemma that allows giving a [[wellordering]] of the relevant mice.

At the level of [[strong cardinalscardinal]]s and above, one constructs an intermediate countably certified core model K^c, and then, if possible, extracts K from K^c.

==Properties of core models==

K_c (and hence K) is a fine-structural countably iterable extender model below long extenders. (It is not currently known how to deal with long extenders, which establish that a cardinal is [[superstrong cardinal|superstrong]].) Here countable iterability means ω₁+1 iterability for all countable elementary substructures of initial segments, and it suffices to develop basic theory, including certain condensation properties. The theory of such models is canonical and well- understood. They satisfy [[Generalized Continuum Hypothesis|GCH]], the [[diamond principle]] for all [[stationary subsetssubset]]s of regular cardinals, the [[square principle]] (except at [[subcompact cardinalscardinal]]s), and other principles holding in L.

K^c is maximal in several senses. K^c computes the successors of measurable and many singular cardinals correctly. Also, it is expected that under an appropriate weakening of countable certifiability, K^c would correctly compute the successors of all [[weakly compact cardinal|weakly compact]] and singular [[strong limit cardinalscardinal]]s correctly. If V is closed under a mouse operator (an inner model operator), then so is K^c. K^c has no sharp: There is no natural non-trivial [[elementary embedding]] of K^c into itself. (However, unlike K, K^c may be elementarily self-embeddable.)

If in addition there are also no Woodin cardinals in this model (except in certain specific cases, it is not known how the core model should be defined if K_c has Woodin cardinals), we can extract the actual core model K. K is also its own core model. K is locally definable and generically absolute: For every generic extension of V, for every cardinal κ>ω₁ in V[G], K as constructed in H(κ) of V[G] equals K∩H(κ). (This would not be possible had K contained Woodin cardinals). K is maximal, universal, and fully iterable. This implies that for every iterable extender model M (called a mouse), there is an elementary embedding M→N and of an initial segment of K into N, and if M is universal, the embedding is of K into M.

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Partial results for the conjecture are that:

#If there is no inner model with a Woodin cardinal, then K^c exists and is fully iterable.

#If (boldface) Σ¹_n+1 [[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^c below κ exists, or there is an ω₁+1 iterable model with measurable limit λ of both Woodin cardinals and cardinals strong up to λ.

#If there is noa innermeasurable modelcardinal withκ, athen strongeither cardinalK^c thatbelow κ exists, or there is aan ω₁+1 iterable model with measurable limit λ of strongboth Woodin cardinals, thenand Kcardinals existsstrong up to λ.

#If every set has a sharp or there is a proper class of subtle cardinals, but there is no inner model with a Woodin cardinal, then K exists.

#If (boldface) Σ¹_n+1 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 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==

{{Reflist}}

* [[W.H. Hugh Woodin]] (June/July 2001). "[httphttps://www.aimathams.org/WWNnotices/coremodel200106/coremodelfea-woodin.pdf The Continuum Hypothesis, Part I]". Notices of the AMS.

* William Mitchell. "Beginning Inner Model Theory" (being Chapter 17 in Volume 3 of "Handbook of Set Theory") at [https://web.archive.org/web/20110617031749/http://www.math.ufl.edu/~wjm/papers/]{{dead link|date=April 2014}}.

* [[Matthew Foreman]] and [[Akihiro Kanamori]] (Editors). "Handbook of Set Theory", Springer Verlag, 2010, ISBN {{isbn|978-1402048432}}.

* Ronald Jensen and John R. Steel. "K without the measurable". Journal of Symbolic Logic Volume 78, Issue 3 (2013), 708-734.

[[Category:Inner model theory]]