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 $|A|\ge \kappa$ there exists a normal measure over $[A{]}^{<\kappa }$, in the following sense.

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

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

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 $(M,R_1,\dots ,R_n)$ with cardinality of the domain $|M|\ge \kappa$, and for every ${\mathrm{\Pi }}_1^1$ sentence $\varphi$ such that $(M,R_1,\dots ,R_n)\models \varphi$, there exists a substructure $(M^{\prime },R_1|M,\dots ,R_n|M)$ with smaller domain (i.e. $|M^{\prime }|<|M|$) that satisfies $\varphi$.

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.

See also

• Indestructibility
• Strongly compact cardinal
• List of large cardinal properties

References

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