# Supercompact cardinal

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

${ }^\lambda M\subseteq M \,.$

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:

$[A]^{< \kappa} := \{x \subseteq A| |x| < \kappa\} \,.$

An ultrafilter U over [A]< κ is fine if it is κ-complete and $\{x \in [A]^{< \kappa}| a \in x\} \in U$, for every $a \in A$. A normal measure over [A]< κ is a fine ultrafilter U over [A]< κ with the additional property that every function $f:[A]^{< \kappa} \to A$ such that $\{x \in [A]^{< \kappa}| f(x)\in x\} \in U$ is constant on a set in $U$. Here "constant on a set in U" means that there is $a \in A$ such that $\{x \in [A]^{< \kappa}| f(x)= a\} \in U$.

## 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 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.