Supercompact cardinal

In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt.

[1] They display a variety of reflection properties.

-supercompact means that there exists an elementary embedding

into a transitive inner model

with critical point

is supercompact means that it is

Alternatively, an uncountable cardinal

there exists a normal measure over

is defined as follows: An ultrafilter

A normal measure over

is a fine ultrafilter

with the additional property that every function

is constant on a set in

Here "constant on a set in

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

, then a cardinal with that property exists below

is supercompact and the generalized continuum hypothesis (GCH) holds below

then it holds everywhere because a bijection between the powerset of

would be a witness of limited rank for the failure of GCH at

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

with cardinality of the domain

, there exists a substructure

with smaller domain (i.e.

[2] Supercompactness has a combinatorial characterization similar to the property of being ineffable.

be the set of all nonempty subsets of

is supercompact iff for every set

(equivalently every cardinal

[3] Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact.