Supercompact cardinals are best motivated as a generalization of [measurable](Measurable.md "Measurable") cardinals, particularly the characterization of measurable cardinals in terms of [elementary embeddings](Elementary%20embedding.md "Elementary embedding") and strong closure properties. The notion of supercompactness and its consequences was initially developed by Solovay and Reinhardt and further elaborated on by Magidor and Gitik, among many others. Assuming the existence of a supercompact is a very strong assumption and the large cardinal strength of supercompact cardinals is seen in a wide (and bewildering) array of set-theoretic contexts, especially the development of strong forcing axioms and establishing regularity properties of sets of reals. The inner model program has yet to reach the level of a supercompact cardinal and this is considered a prominent open problem in the program itself. Curiously, by results of Woodin, should the inner program reach the level of a supercompact, there is a sense in which it will have reached all greater large cardinals, a startling contrast to previous advances in the program.
## Formal definition and equivalent characterizations
### Elementary embedding definition
Generalizing the [elementary embedding](Elementary%20embedding.md "Elementary embedding") characterization of measurable cardinal, a cardinal $\kappa$ is *$\theta$-supercompact* if there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$, i.e. $M$ is closed under arbitrary sequences of length $\theta$.
Under the <a href="Axiom_of_choice" class="mw-redirect" title="Axiom of choice">axiom of choice</a>, one may assume without loss of generality that $j(\kappa)\gt\theta$.
$\kappa$ is then said to be *supercompact* if it is $\theta$-supercompact for all $\theta$.
>[!note] Remark
> $\kappa$ is said to be *supercompact* if for every $\theta$ there is an elementary embedding $j:V\to M$ with $M$ a transitive class, such that $j$ has critical point $\kappa$ and $M^\theta\subset M$.
It is worth noting that, using this formulation, $H_{\theta^+}$ must be contained in the transitive class $M$.
### Ultrafilter definition
There is an alternative formulation that is expressible in $\text{ZFC}$ using certain [[Filter |ultrafilters]] with somewhat technical properties:
For $\theta\geq\kappa$, $\kappa$ if $\theta$-supercompact if there is a [[Filter#Normal and fine filters|normal and fine measure]] on $\mathcal{P}_\kappa(\theta)$.
$\kappa$ is supercompact if for every set $A$ with $|A|\geq\kappa$, there is a normal fine measure on $\mathcal{P}_\kappa(A)$.
One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [seed](Seed.md "Seed") $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $\mathcal{P}_\kappa(\theta)$.
Conversely, the ultrapower by a normal fine measure $U$ on $\mathcal{P}_\kappa(\theta)$ gives rise to an embedding $j:V\to M$ (here $M$ is identified with the transitive collapse of the ultrapower by $U$). It is then straightforward to check that $\theta$ is the critical point of this embedding and that $M$ is sufficiently closed, thus witnessing $\theta$-supercompactness of $\kappa$.
### Elementary embeddings of initial segments definition
A third characterization was given by Magidor \[Mag71\] in terms of elementary embeddings from initial segments of $V$ into other (larger) initial segments of $V$, but in this characterization, the supercompact cardinal $\kappa$ is the *image* of the critical point of this embedding, rather than the critical point itself:
$\kappa$ is supercompact if and only if for every $\eta>\kappa$ there is $\alpha<\kappa$ such that there exists a nontrivial elementary embedding $j:V_\alpha\to V_\eta$ such that $j(\mathrm{crit}(j))=\kappa$.
([Remarkable](Remarkable.md "Remarkable") cardinals could be called virtually supercompact, because one of their definitions is an exact analogue of this one (with forcing extension)){% cite Gitmana %}
## Properties
If $\kappa$ is supercompact, then there are $2^{2^\kappa}$ [normal fine measures](Filter.md "Filter")
on $\kappa$, also for every $\lambda\geq\kappa$ there are
$2^{2^{\lambda^{<\kappa}}}$ normal fine measures on
$\mathcal{P}_\kappa(\lambda)$.
Every supercompact has
<a href="Mitchell_order" class="mw-redirect" title="Mitchell order">Mitchell order</a>
$(2^\kappa)^+\geq\kappa^{++}$.
If $\kappa$ is $\lambda$-supercompact then it is also
$\mu$-supercompact for every $\mu<\lambda$. If
$\lambda\geq\kappa$ is regular, $\kappa$ is $\lambda$-supercompact,
then every $\alpha<\kappa$ that is $\gamma$-supercompact for all
$\gamma<\kappa$ (if any exists) is also $\lambda$-supercompact. In
the same vein, for every cardinals $\kappa<\lambda$, if $\lambda$
is supercompact and $\kappa$ is $\gamma$-supercompact for all
$\gamma<\lambda$, then $\kappa$ is also supercompact.
### Laver's diamond function
*Laver's theorem* asserts that if $\kappa$ is supercompact, there exists a function $f:\kappa\to V_\kappa$ such that for every $x$ and $\lambda\geq\kappa$ with $\|tc(x)\|\leq\lambda$ there exists a normal fine measure $U$ on $\mathcal{P}_\kappa(\lambda)$ such that $j_U(f)(\kappa)=x$, where $j_U$ is the elementary embedding generated from $U$. $f$ is called a *Laver function*.
Here $tc(x)$ is the *transitive closure* of $x$ (i.e. the smallest transitive set containing $x$).
## Supercompact cardinals and forcing
### The continuum hypothesis and supercompact cardinals
If $\kappa$ is $\lambda$-supercompact and $2^\alpha=\alpha^{+}$ for
every $\alpha<\kappa$, then $2^\alpha=\alpha^{+}$ for every
$\alpha\leq\lambda$. Consequently, if the
<a href="GCH" class="mw-redirect" title="GCH">generalized continuum hypothesis</a>
holds below a supercompact cardinal, then it holds everywhere.
The existence of a supercompact implies the consistency of the failure
of the *singular cardinal hypothesis*, i.e. it is consistent that the
generalized continuum hypothesis fails at a strong limit singular
cardinal. It also implies the consistency of the failure of the
$\text{GCH}$ at a measurable cardinal.
By combining results of Magidor, Shelah and Gitik, one can show that the
existence of a supercompact also implies the existence of a [](Forcing.md)
in which $2^{\aleph_\alpha}<\aleph_{\omega_1}$ for all
$\alpha<\omega_1$, but also
$2^{\aleph_{\omega_1}}>\aleph_{\omega_1+\alpha+1}$ for any
prescribed $\alpha<\omega_2$. Similarly, one can have a generic
extension in which the $\text{GCH}$ holds below $\aleph_\omega$ but
$2^{\aleph_\omega}>\aleph_{\omega+\alpha+1}$ for any
prescribed $\alpha<\omega_1$.
Woodin and Cummings furthermore showed that if there exists a
supercompact, then there is a generic extension in which
$2^\kappa=\kappa^{++}$ for *every* cardinal $\kappa$, i.e. the
$\text{GCH}$ fails *everywhere*(!).
The
<a href="Ultrapower_axiom" class="mw-redirect" title="Ultrapower axiom">ultrapower axiom</a>,
if consistent with a supercompact, implies that the $\text{GCH}$ holds
above the least supercompact.
### Laver preparation
*Indestructibility, including the Laver diamond.*
### Proper forcing axiom
Baumgartner proved that if there is a supercompact cardinal, then the
<a href="Proper_forcing_axiom" class="mw-redirect" title="Proper forcing axiom">proper forcing axiom</a>
holds in a
[forcing](Forcing.md "Forcing")
extension. $\text{PFA}