In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
If λ is any ordinal, κ is λsupercompact 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
λ
M
⊆
M
.
That is, M contains all of its λsequences. Then κ is supercompact means that it is λsupercompact 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
]
<
κ
:=
{
x
⊆
A


x

<
κ
}
.
An ultrafilter U over [A]^{< κ} is fine if it is κcomplete and
{
x
∈
[
A
]
<
κ

a
∈
x
}
∈
U
, for every
a
∈
A
. A normal measure over [A]^{< κ} is a fine ultrafilter U over [A]^{< κ} with the additional property that every function
f
:
[
A
]
<
κ
→
A
such that
{
x
∈
[
A
]
<
κ

f
(
x
)
∈
x
}
∈
U
is constant on a set in
U
. Here "constant on a set in U" means that there is
a
∈
A
such that
{
x
∈
[
A
]
<
κ

f
(
x
)
=
a
}
∈
U
.
Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3huge 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.