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

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

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

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

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

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

An ultrafilter ${\displaystyle U}$ over ${\displaystyle [A]^{<\kappa }}$ is fine if it is ${\displaystyle \kappa }$-complete and ${\displaystyle \{x\in [A]^{<\kappa }\mid a\in x\}\in U}$, for every ${\displaystyle a\in A}$. A normal measure over ${\displaystyle [A]^{<\kappa }}$ is a fine ultrafilter ${\displaystyle U}$ over ${\displaystyle [A]^{<\kappa }}$ with the additional property that every function ${\displaystyle f:[A]^{<\kappa }\to A}$ such that ${\displaystyle \{x\in [A]^{<\kappa }|f(x)\in x\}\in U}$ is constant on a set in ${\displaystyle U}$. Here "constant on a set in ${\displaystyle U}$" means that there is ${\displaystyle a\in A}$ such that ${\displaystyle \{x\in [A]^{<\kappa }|f(x)=a\}\in U}$.

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 ${\displaystyle \kappa }$, then a cardinal with that property exists below ${\displaystyle \kappa }$. For example, if ${\displaystyle \kappa }$ is supercompact and the generalized continuum hypothesis (GCH) holds below ${\displaystyle \kappa }$ then it holds everywhere because a bijection between the powerset of ${\displaystyle \nu }$ and a cardinal at least ${\displaystyle \nu ^{++}}$ would be a witness of limited rank for the failure of GCH at ${\displaystyle \nu }$ so it would also have to exist below ${\displaystyle \nu }$.

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 ${\displaystyle \kappa }$ such that for every structure ${\displaystyle (M,R_{1},\ldots ,R_{n})}$ with cardinality of the domain ${\displaystyle \vert M\vert \geq \kappa }$, and for every ${\displaystyle \Pi _{1}^{1}}$ sentence ${\displaystyle \phi }$ such that ${\displaystyle (M,R_{1},\ldots ,R_{n})\vDash \phi }$, there exists a substructure ${\displaystyle (M',R_{1}\vert M,\ldots ,R_{n}\vert M)}$ with smaller domain (i.e. ${\displaystyle \vert M'\vert <\vert M\vert }$) that satisfies ${\displaystyle \phi }$.[2]

Supercompactness has a combinatorial characterization similar to the property of being ineffable. Let ${\displaystyle P_{\kappa }(A)}$ be the set of all nonempty subsets of ${\displaystyle A}$ which have cardinality ${\displaystyle <\kappa }$. A cardinal ${\displaystyle \kappa }$ is supercompact iff for every set ${\displaystyle A}$ (equivalently every cardinal ${\displaystyle \alpha }$), for every function ${\displaystyle f:P_{\kappa }(A)\to P_{\kappa }(A)}$, if ${\displaystyle f(X)\subseteq X}$ for all ${\displaystyle X\in P_{\kappa }(A)}$, then there is some ${\displaystyle B\subseteq A}$ such that ${\displaystyle \{X\mid f(X)=B\cap X\}}$ is stationary.[3]

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

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.

Citations

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.