Supercompact cardinal

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.

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


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

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

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

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


 * $$[A]^{<\kappa} := \{x \subseteq A\mid \vert x\vert < \kappa\}$$.

An ultrafilter $$U$$ over $$[A]^{<\kappa}$$ is fine if it is $$\kappa$$-complete and $$\{x \in [A]^{<\kappa}\mid a \in x\} \in U$$, for every $$a \in A$$. A normal measure over $$[A]^{<\kappa}$$ is a fine ultrafilter $$U$$ over $$[A]^{<\kappa}$$ 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 $$\kappa$$, then a cardinal with that property exists below $$\kappa$$. For example, if $$\kappa$$ is supercompact and the generalized continuum hypothesis (GCH) holds below $$\kappa$$ then it holds everywhere because a bijection between the powerset of $$\nu$$ and a cardinal at least $$\nu^{++}$$ would be a witness of limited rank for the failure of GCH at $$\nu$$ so it would also have to exist below $$\nu$$.

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

The least supercompact cardinal is the least $$\kappa$$ such that for every structure $$(M,R_1,\ldots,R_n)$$ with cardinality of the domain $$\vert M\vert\geq\kappa$$, and for every $$\Pi_1^1$$ sentence $$\phi$$ such that $$(M,R_1,\ldots,R_n)\vDash\phi$$, there exists a substructure $$(M',R_1\vert M,\ldots,R_n\vert M)$$ with smaller domain (i.e. $$\vert M'\vert<\vert M\vert$$) that satisfies $$\phi$$.

Supercompactness has a combinatorial characterization similar to the property of being ineffable. Let $$P_\kappa(A)$$ be the set of all nonempty subsets of $$A$$ which have cardinality $$<\kappa$$. A cardinal $$\kappa$$ is supercompact iff for every set $$A$$ (equivalently every cardinal $$\alpha$$), for every function $$f:P_\kappa(A)\to P_\kappa(A)$$, if $$f(X)\subseteq X$$ for all $$X\in P_\kappa(A)$$, then there is some $$B\subseteq A$$ such that $$\{X\mid f(X)=B\cap X\}$$ is stationary.

Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact.