Core model
Encyclopedia
In set theory
, the core model is a definable inner model
of the universe
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
) 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.
'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
s and iteration trees to construct a core model below a Woodin cardinal.
. An important ingredient of the construction is the comparison lemma that allows giving a wellordering of the relevant mice.
At the level of strong cardinals and above, one constructs an intermediate countably certified core model Kc, and then, if possible, extracts K from Kc.
, the diamond principle for all stationary subsets of regular cardinals, the square principle (except at subcompact cardinals), and other principles holding in L.
Kc is maximal in several senses. Kc computes the successors of measurable and many singular cardinals correctly. Also, it is expected that under an appropriate weakening of countable certifiability, Kc 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 Kc. Kc has no sharp: There is no natural non-trivial elementary embedding of Kc into itself. (However, unlike K, Kc 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 Kc 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 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→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 Kc generalizes to K(X) and Kc(X).
Partial results for the conjecture are that:
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.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the core model is a definable inner model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
of the universe
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
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
Covering lemma
In mathematics, under various anti-large cardinal assumptions, one can prove the existence of the canonical inner model, called the Core Model, that is, in a sense, maximal and approximates the structure of V...
. Intuitively, the core model is "the largest canonical inner model there is" (Ernest Schimmerling and John R. Steel
John R. Steel
John Robert Steel is a set theorist at University of California, Berkeley . He has made many contributions to the theory of inner models and determinacy. With Donald A. Martin, he proved projective determinacy, assuming the existence of sufficient large cardinals. He earned his Ph.D...
) 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 GödelKurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
's constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
L. Ronald Jensen
Ronald Jensen
Ronald Björn Jensen is an American mathematician active in Europe, primarily known for his work in mathematical logic and set theory.-Career:...
proved the covering lemma
Covering lemma
In mathematics, under various anti-large cardinal assumptions, one can prove the existence of the canonical inner model, called the Core Model, that is, in a sense, maximal and approximates the structure of V...
for L in the 1970s under the assumption of the non-existence of zero sharp
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
, establishing that L is the "core model below zero sharp". The work of Solovay isolated another core model L[U], for U an ultrafilter
Ultrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
on a measurable cardinal
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
(and its associated "sharp", zero dagger
Zero dagger
In set theory, 0† is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions...
). 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
Extender (set theory)
In set theory, an extender is a set which represents an elementary embedding having large cardinal properties. A nonprincipal ultrafilter is the most basic case of an 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 miceMouse (set theory)
In set theory, a mouse is a small model of Zermelo-Fraenkel set theory with desirable properties. The exact definition depends on the context. In most cases, there is a technical definition of "premouse" and an added condition of iterability : a mouse is then an iterable premouse...
. An important ingredient of the construction is the comparison lemma that allows giving a wellordering of the relevant mice.
At the level of strong cardinals and above, one constructs an intermediate countably certified core model Kc, and then, if possible, extracts K from Kc.
Properties of core models
Kc (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 GCHGCH
GCH may refer to:* The Generalized Continuum Hypothesis in mathematics * Knight Grand Cross of the Royal Guelphic Order* Gas central heating* Gym Class Heroes...
, the diamond principle for all stationary subsets of regular cardinals, the square principle (except at subcompact cardinals), and other principles holding in L.
Kc is maximal in several senses. Kc computes the successors of measurable and many singular cardinals correctly. Also, it is expected that under an appropriate weakening of countable certifiability, Kc 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 Kc. Kc has no sharp: There is no natural non-trivial elementary embedding of Kc into itself. (However, unlike K, Kc 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 Kc 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 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→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 Kc generalizes to K(X) and Kc(X).
Construction of core models
Conjecture:- If there is no ω1+1 iterable model with long extenders (and hence models with superstrong cardinals), then Kc exists.
- If Kc 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 Kc exists and is fully iterable.
- If there is a measurable cardinal κ, then either Kc 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.
- If (boldface) Σ1n+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 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.