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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]]s. 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.

{{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 [[setSet (mathematics)|sets]]s. 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.

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]].

==History==

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 to give a [[wellordering]] of the relevant mice.

The first core model was [[Kurt Gödel]]'s [[constructible universe]] '''L'''. [[Ronald Jensen]] proved the [[Coveringcovering Lemmalemma]] 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. Larger core models include the Mitchell core model and thea Steelgeneralized corecovering modellemma below afor '''L'''[[Woodin cardinal]''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]].

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

==Construction of core models==

<b>Conjecture:</b>

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 to givegiving a [[wellordering]] of the relevant mice.

*If there is no ω₁+1 iterable model with long extenders (and hence models with superstrong cardinals), then K^c exists.<br/>

*If K^c exists and as constructed in every generic extension of V (equivalently, under some generic collapse Coll(ω, &lt;κ) for a sufficiently large ordinal κ) satisfies "there are no Woodin cardinals", then the Core Model K exists.

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.

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 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 no inner model with a strong cardinal that is a limit of strong cardinals, then K exists.

#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. <br/>

#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.

==Properties of core models==

K_c (and hence K) is a finestructuralfine-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.)

K_c (and hence K) is a finestructural countably iterable extender model below long extenders. 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 GCH, the diamond principle for all stationary subsets of regular cardinals, the square principle (except at subcompact cardinals), and other principles holding in L.

LetIf Kin beaddition athere coreare model withalso no Woodin cardinals orin long extenders.this model (Exceptexcept in certain specific cases, it is not known how the core model should be defined if K_c has Woodin cardinals), orwe howcan toextract dealthe withactual longcore extenders.)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 interesect HK∩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&rarr;NM→N and of an initial segment of K into N, and if M is universal, the embedding is of K into M.

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 and singular strong limit cardinals 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.)

It is conjectured that if K exists and V is closed under a sharp operator M, then K is Σ¹₁ correct allowing real numbers in K as parameters, and M as a predicate. That amounts to Σ¹₃ correctness (in the usual sense) if M is x→x^#.

Let K be a core model with no Woodin cardinals or long extenders. (Except in certain specific cases, it is not known how the core model should be defined if K_c has Woodin cardinals, or how to deal with long extenders.) 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 interesect 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 mouse), there is an elementary embedding M&rarr;N and of an initial segment of K into N, and if M is universal, the embedding is of K into M.

The core model can also be defined above a particular set of ordinals X: X belongbelongs to K(X), but K(X) satisfies the usual properties of K above X. If there is no iterable inner model with ω Woodin cardinals, then for some X, K(X) exists. The above discussion of K and K^c generalizes to K(X) and K^c(X).

It is conjectured that if K exists and V is closed under a sharp operator M, then K is Σ¹₁ correct allowing real numbers in K as parameters, and M as a predicate. That amounts to Σ¹₃ correctness if M is x→x^#.

==Construction of core models==

<b>'''Conjecture:</b>'''

*If there is no ω₁+1 iterable model with long extenders (and hence models with superstrong cardinals), then K^c exists.<br/>

*If K^c exists and as constructed in every generic extension of V (equivalently, under some generic collapse Coll(ω, &lt;<κ) for a sufficiently large ordinal κ) satisfies "there are no Woodin cardinals", then the Core Model K exists.

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 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.

The core model can also be defined above a particular set of ordinals X: X belong K(X), but K(X) satisfies the usual properties of K above X. If there is no iterable inner model with ω Woodin cardinals, then for some X, K(X) exists. The above discussion of K and K^c generalizes to K(X) and K^c(X).

==References==

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 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.

{{Reflist}}

* [[W. Hugh Woodin]] (June/July 2001). "[https://www.ams.org/notices/200106/fea-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/].

* [[Matthew Foreman]] and [[Akihiro Kanamori]] (Editors). "Handbook of Set Theory", Springer Verlag, 2010, {{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]]

[[Category:Large cardinals]]