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.

See also

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

References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
• Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3.