Supercompact cardinal

{{Short description|Large cardinal from set theory}} In set theory, a '''supercompact cardinal''' is a type of large cardinal independently introduced by Solovay and Reinhardt.<ref>A. Kanamori, "[https://math.bu.edu/people/aki/19.pdf Kunen and set theory]", pp.2450--2451. Topology and its Applications, vol. 158 (2011).</ref> They display a variety of reflection properties.

==Formal definition==

If <math>\lambda</math> is any ordinal, <math>\kappa</math> is '''<math>\lambda</math>-supercompact''' means that there exists an elementary embedding <math>j</math> from the universe <math>V</math> into a transitive inner model <math>M</math> with critical point <math>\kappa</math>, <math>j(\kappa)>\lambda</math> and

:<math>{ }^\lambda M\subseteq M \,.</math>

That is, <math>M</math> contains all of its <math>\lambda</math>-sequences. Then <math>\kappa</math> is '''supercompact''' means that it is <math>\lambda</math>-supercompact for all ordinals <math>\lambda</math>.

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

<math>[A]^{<\kappa}</math> is defined as follows:

:<math>[A]^{<\kappa} := \{x \subseteq A\mid \vert x\vert < \kappa\}</math>.

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

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

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 <math>\kappa</math> such that for every structure <math>(M,R_1,\ldots,R_n)</math> with cardinality of the domain <math>\vert M\vert\geq\kappa</math>, and for every <math>\Pi_1^1</math> sentence <math>\phi</math> such that <math>(M,R_1,\ldots,R_n)\vDash\phi</math>, there exists a substructure <math>(M',R_1\vert M,\ldots,R_n\vert M)</math> with smaller domain (i.e. <math>\vert M'\vert<\vert M\vert</math>) that satisfies <math>\phi</math>.<ref>{{cite journal | last1=Magidor | first1=M. | authorlink1=Menachem Magidor | title=On the Role of Supercompact and Extendible Cardinals in Logic | date=1971 | pages=147–157 | journal=Israel Journal of Mathematics | volume=10 | issue=2 | doi=10.1007/BF02771565 | doi-access=}}</ref>

Supercompactness has a combinatorial characterization similar to the property of being ineffable. Let <math>P_\kappa(A)</math> be the set of all nonempty subsets of <math>A</math> which have cardinality <math><\kappa</math>. A cardinal <math>\kappa</math> is supercompact iff for every set <math>A</math> (equivalently every cardinal <math>\alpha</math>), for every function <math>f:P_\kappa(A)\to P_\kappa(A)</math>, if <math>f(X)\subseteq X</math> for all <math>X\in P_\kappa(A)</math>, then there is some <math>B\subseteq A</math> such that <math>\{X\mid f(X)=B\cap X\}</math> is stationary (in \(P_\kappa(A)\)).<ref>M. Magidor, [https://www.ams.org/journals/proc/1974-042-01/S0002-9939-1974-0327518-9/S0002-9939-1974-0327518-9.pdf Combinatorial Characterization of Supercompact Cardinals], pp.281--282. Proceedings of the American Mathematical Society, vol. 42 no. 1, 1974.</ref>

Magidor obtained a variant of the tree property which holds for an inaccessible cardinal iff it is supercompact.<ref>S. Hachtman, S. Sinapova, "[https://sites.math.rutgers.edu/~ds2005/ITP.pdf The super tree property at the successor of a singular]". Israel Journal of Mathematics, vol 236, iss. 1 (2020), pp.473--500.</ref>

==References== {{refbegin}} * {{cite book|author=Drake, F. R.|title=Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76)|publisher=Elsevier Science Ltd|year=1974|isbn=0-444-10535-2}} * {{cite book|author=Jech, Thomas|title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=3-540-44085-2|authorlink=Thomas Jech}} * {{cite book|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|title-link= The Higher Infinite |edition=2nd|isbn=3-540-00384-3}} {{refend}}

===Citations=== {{reflist}}

Category:Large cardinals