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 , in the following sense.

is defined as follows:

: := \.

An ultrafilter $U$ over is ''fine'' if it is $\kappa$-complete and $\ \in U$, for every $a \in A$. A normal measure over is a fine ultrafilter $U$ over with the additional property that every function f: \to A such that $\ \in U$ is constant on a set in $U$. Here "constant on a set in $U$" means that there is $a \in A$ such that $\ \in U$.

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 $\$ is stationary.

Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact.S. Hachtman, S. Sinapova,
The super tree property at the successor of a singular
. Israel Journal of Mathematics, vol 236, iss. 1 (2020), pp.473--500.

See also

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

References

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Citations

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Large cardinals