set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, the core model is a definable

inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangle be ...

of the

universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...

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 John Robert Steel (born October 30, 1948) is an American set theory, set theorist at University of California, Berkeley (formerly at University of California, Los Angeles, UCLA). He has made many contributions to the theory of inner models and de ...

) and is typically associated with a

large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least � ...

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 Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

's constructible universe L. Ronald Jensen proved the

covering lemma In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinals leads to the existence of a canonical inner model, called the core model, that is, in a sense, maximal and approximates the struc ...

for L in the 1970s under the assumption of the non-existence of

zero sharp In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel nu ...

, 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 In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The def ...

). 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 In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions :f : \lambda \to \lambda there exists a cardinal \kappa < \lambda with :

\ \subseteq \kappa

and an

Construction of core models

Core models are constructed by transfinite recursion from small fragments of the core model called

mice A mouse ( : mice) is a small rodent. Characteristically, mice are known to have a pointed snout, small rounded ears, a body-length scaly tail, and a high breeding rate. The best known mouse species is the common house mouse (''Mus musculus' ...

. An important ingredient of the construction is the comparison lemma that allows giving a

wellordering In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...

of the relevant mice. 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.

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.) 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 In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe () and that implies the continuum hypothesis. Jensen extracted t ...

for all

stationary subset In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closel ...

s of regular cardinals, the

square principle In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a ...

(except at

subcompact cardinal In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number ''κ'' is subcompact if and only if for every ''A'' ⊂ ''H''(''κ''+) there is a non-trivial elementary embedding j:(''H''(''μ''+), ''B' ...

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 and singular

strong limit cardinal In mathematics, limit cardinals are certain cardinal numbers. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated success ...

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 K as constructed in H(κ) of V 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. 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^#. The core model can also be defined above a particular set of ordinals X: X belongs 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).

Construction of core models

Conjecture: *If there is no ω₁+1 iterable model with long extenders (and hence models with superstrong cardinals), then K^c exists. *If K^c exists and as constructed in every generic extension of V (equivalently, under some generic collapse Coll(ω, <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 exists. #If (boldface) Σ¹_n 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 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

W. Hugh Woodin William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...

(June/July 2001)

Notices of the AMS. * William Mitchell. "Beginning Inner Model Theory" (being Chapter 17 in Volume 3 of "Handbook of Set Theory") a

*

Matthew Foreman Matthew Dean Foreman is an American mathematician at University of California, Irvine. He has made notable contributions in set theory and in ergodic theory. Biography Born in Los Alamos, New Mexico, Foreman earned his Ph.D. from the Univer ...

and

Akihiro Kanamori is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinal property, large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especia ...

(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. Inner model theory Large cardinals