Supercompact cardinal

In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt.^[1] 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 $\phi$ .^[2]

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.^[3]

Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact.^[4]

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.

Citations

^ A. Kanamori, "Kunen and set theory", pp.2450--2451. Topology and its Applications, vol. 158 (2011).
^ Magidor, M. (1971). "On the Role of Supercompact and Extendible Cardinals in Logic". Israel Journal of Mathematics. 10 (2): 147–157. doi:10.1007/BF02771565.
^ M. Magidor, Combinatorial Characterization of Supercompact Cardinals, pp.281--282. Proceedings of the American Mathematical Society, vol. 42 no. 1, 1974.
^ 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.

[1] A. Kanamori, "Kunen and set theory", pp.2450--2451. Topology and its Applications, vol. 158 (2011).

[2] Magidor, M. (1971). "On the Role of Supercompact and Extendible Cardinals in Logic". Israel Journal of Mathematics. 10 (2): 147–157. doi:10.1007/BF02771565.

[3] M. Magidor, Combinatorial Characterization of Supercompact Cardinals, pp.281--282. Proceedings of the American Mathematical Society, vol. 42 no. 1, 1974.

[4] 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.

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