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Supercompact cardinal - Wikipedia

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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.

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}.

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]

  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.