In set theory, a **supercompact cardinal** is a type of large cardinal. They display a variety of reflection properties.

## Formal definition

If *λ* is any ordinal, *κ* is ** λ-supercompact** means that there exists an elementary embedding

*j*from the universe

*V*into a transitive inner model

*M*with critical point

*κ*,

*j*(

*κ*)>

*λ*and

That is, *M* contains all of its *λ*-sequences. Then *κ* is **supercompact** means that it is *λ*-supercompact for all ordinals *λ*.

Alternatively, an uncountable cardinal *κ* is **supercompact** if for every *A* such that |*A*| ≥ *κ* there exists a normal measure over [*A*]^{< κ}, in the following sense.

[*A*]^{< κ} is defined as follows:

An ultrafilter *U* over [*A*]^{< κ} is *fine* if it is *κ*-complete and , for every . A normal measure over [*A*]^{< κ} is a fine ultrafilter *U* over [*A*]^{< κ} with the additional property that every function such that is constant on a set in . Here "constant on a set in *U*" means that there is 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 *κ*, then a cardinal with that property exists below κ. For example, if *κ* is supercompact and the generalized continuum hypothesis (GCH) holds below *κ* then it holds everywhere because a bijection between the powerset of *ν* and a cardinal at least *ν*^{++} would be a witness of limited rank for the failure of GCH at *ν* so it would also have to exist below *κ*.

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

## See also

## References

- Drake, F. R. (1974).
*Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76)*. Elsevier Science Ltd. ISBN 0-444-10535-2. - Jech, Thomas (2002).
*Set theory, third millennium edition (revised and expanded)*. Springer. ISBN 3-540-44085-2. - Kanamori, Akihiro (2003).
*The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings*(2nd ed.). Springer. ISBN 3-540-00384-3.