Core model: Difference between revisions

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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.

==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''].

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==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 [[GCH]], the [[diamond principle]] for all stationary subsets of regular cardinals, the square principle (except at subcompact cardinals), 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 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.)

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==References==

* W.H. Woodin (2001). [http://www.aimath.org/WWN/coremodel/coremodel.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 [http://www.math.ufl.edu/~wjm/papers/]{{brokendead link|date=April 2014}}.

* Matthew Foreman and Akihiro Kanamori (Editors). "Handbook of Set Theory", Springer Verlag, 2010, ISBN 978-1402048432.

[[Category:Inner model theory]]