Supercompact cardinal explained

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

λ is any ordinal, \kappa is λ -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)> λ and{ } ^{λ} M\subseteq M .

That is,

M contains all of its λ -sequences. Then \kappa is supercompact means that it is λ -supercompact for all ordinals λ .

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\} .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] ^{<} | 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]

See also

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

References

• Book: Drake, F. R.. Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. 1974. 0-444-10535-2.
• Book: Thomas Jech

. Jech, Thomas. Set theory, third millennium edition (revised and expanded). Springer. 2002. 3-540-44085-2. Thomas Jech.

• Book: Kanamori, Akihiro. Akihiro Kanamori

. Akihiro Kanamori. 2003. Springer. The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. The Higher Infinite . 2nd. 3-540-00384-3.

Citations

Notes and References

1. A. Kanamori, "Kunen and set theory", pp.2450--2451. Topology and its Applications, vol. 158 (2011).
2. Magidor . M. . Menachem Magidor. On the Role of Supercompact and Extendible Cardinals in Logic. 1971. 147–157. Israel Journal of Mathematics. 10. 2. 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.