In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties.

Formal definition

If \lambda is any ordinal, \kappa is \lambda-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point \kappa, and That is, M contains all of its \lambda-sequences. Then \kappa is supercompact means that it is \lambda-supercompact for all ordinals \lambda. Alternatively, an uncountable cardinal \kappa is supercompact if for every A such that there exists a normal measure over, in the following sense. is defined as follows: An ultrafilter U over is fine if it is \kappa-complete and, for every a \in A. A normal measure over is a fine ultrafilter U over with the additional property that every function such that is constant on a set in U. Here "constant on a set in U" means that there is a \in A such that.

Properties

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal \kappa, then a cardinal with that property exists below \kappa. For example, if \kappa is supercompact and the generalized continuum hypothesis (GCH) holds below \kappa then it holds everywhere because a bijection between the powerset of \nu and a cardinal at least \nu^{++} would be a witness of limited rank for the failure of GCH at \nu so it would also have to exist below \nu. Finding a canonical inner model for supercompact cardinals is one of the major problems of inner model theory. The least supercompact cardinal is the least \kappa such that for every structure with cardinality of the domain, and for every \Pi_1^1 sentence \phi such that , there exists a substructure with smaller domain (i.e. ) that satisfies \phi. Supercompactness has a combinatorial characterization similar to the property of being ineffable. Let P_\kappa(A) be the set of all nonempty subsets of A which have cardinality <\kappa. A cardinal \kappa is supercompact iff for every set A (equivalently every cardinal \alpha), for every function, if for all , then there is some such that is stationary. Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact.

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