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

is any ordinal is -supercompact means that there exists an elementary embedding into a transitive inner model with critical point and

is supercompact.

is supercompact there exists a normal measure, in the following sense.

is defined as follows:

1. .

An ultrafilter is fine.

Properties

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal is supercompact and the generalized continuum hypothesis.

Finding a canonical inner model for supercompact cardinals is one of the major problems of inner model theory.

.

Supercompactness has a combinatorial characterization similar to the property of being ineffable is stationary (in \(P_\kappa(A)\)).

Magidor obtained a variant of the tree property which holds for an inaccessible cardinal if it is supercompact.

See also

• Indestructibility

• Strongly compact cardinal

• List of large cardinal properties

References

Citations