Supercompact cardinal

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, j(\kappa)>\lambda and :{ }^\lambda M\subseteq M \,. 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 \vert A\vert\geq\kappa there exists a normal measure over [A]^{, in the following sense. [A]^{ is defined as follows: :[A]^{. An ultrafilter U over [A]^{ is fine if it is \kappa-complete and \{x \in [A]^{, for every a \in A. A normal measure over [A]^{ is a fine ultrafilter U over [A]^{ with the additional property that every function f:[A]^{ such that \{x \in [A]^{ is constant on a set in U. Here "constant on a set in U" means that there is a \in A such that \{x \in [A]^{. ==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 (M,R_1,\ldots,R_n) with cardinality of the domain \vert M\vert\geq\kappa, and for every \Pi_1^1 sentence \phi such that (M,R_1,\ldots,R_n)\vDash\phi, there exists a substructure (M',R_1\vert M,\ldots,R_n\vert M) with smaller domain (i.e. \vert M'\vert) that satisfies \phi. Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact. ==See also==