# 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

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

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:

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

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

## 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.