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.

If ${\displaystyle \lambda }$ is any ordinal, ${\displaystyle \kappa }$ is ${\displaystyle \lambda }$-supercompact means that there exists an elementary embedding ${\displaystyle j}$ from the universe ${\displaystyle V}$ into a transitive inner model ${\displaystyle M}$ with critical point ${\displaystyle \kappa }$, ${\displaystyle j(\kappa )>\lambda }$ and

That is, ${\displaystyle M}$ contains all of its ${\displaystyle \lambda }$-sequences. Then ${\displaystyle \kappa }$ is supercompact means that it is ${\displaystyle \lambda }$-supercompact for all ordinals ${\displaystyle \lambda }$.

Alternatively, an uncountable cardinal ${\displaystyle \kappa }$ is supercompact if for every ${\displaystyle A}$ such that ${\displaystyle \vert A\vert \geq \kappa }$ there exists a normal measure over ${\displaystyle [A]^{<\kappa }}$, in the following sense.

${\displaystyle [A]^{<\kappa }}$ is defined as follows:

An ultrafilter ${\displaystyle U}$ over ${\displaystyle [A]^{<\kappa }}$ is fine if it is ${\displaystyle \kappa }$-complete and ${\displaystyle \{x\in [A]^{<\kappa }\mid a\in x\}\in U}$, for every ${\displaystyle a\in A}$. A normal measure over ${\displaystyle [A]^{<\kappa }}$ is a fine ultrafilter ${\displaystyle U}$ over ${\displaystyle [A]^{<\kappa }}$ 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 ${\displaystyle U}$" means that there is ${\displaystyle a\in A}$ such that ${\displaystyle \{x\in [A]^{<\kappa }|f(x)=a\}\in U}$.

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 ${\displaystyle \kappa }$, then a cardinal with that property exists below ${\displaystyle \kappa }$. For example, if ${\displaystyle \kappa }$ is supercompact and the generalized continuum hypothesis (GCH) holds below ${\displaystyle \kappa }$ then it holds everywhere because a bijection between the powerset of ${\displaystyle \nu }$ and a cardinal at least ${\displaystyle \nu ^{++}}$ would be a witness of limited rank for the failure of GCH at ${\displaystyle \nu }$ so it would also have to exist below ${\displaystyle \nu }$.

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