Supercompact cardinal

In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties.

Formal definition

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

[math]\displaystyle{ { }^\lambda M\subseteq M \,. }[/math]

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

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

[math]\displaystyle{ [A]^{\lt \kappa} }[/math] is defined as follows:

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

An ultrafilter [math]\displaystyle{ U }[/math] over [math]\displaystyle{ [A]^{\lt \kappa} }[/math] is fine if it is [math]\displaystyle{ \kappa }[/math]-complete and [math]\displaystyle{ \{x \in [A]^{\lt \kappa}\mid a \in x\} \in U }[/math], for every [math]\displaystyle{ a \in A }[/math]. A normal measure over [math]\displaystyle{ [A]^{\lt \kappa} }[/math] is a fine ultrafilter [math]\displaystyle{ U }[/math] over [math]\displaystyle{ [A]^{\lt \kappa} }[/math] with the additional property that every function [math]\displaystyle{ f:[A]^{\lt \kappa} \to A }[/math] such that [math]\displaystyle{ \{x \in [A]^{\lt \kappa}| f(x)\in x\} \in U }[/math] is constant on a set in [math]\displaystyle{ U }[/math]. Here "constant on a set in [math]\displaystyle{ U }[/math]" means that there is [math]\displaystyle{ a \in A }[/math] such that [math]\displaystyle{ \{x \in [A]^{\lt \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]\displaystyle{ \kappa }[/math], then a cardinal with that property exists below [math]\displaystyle{ \kappa }[/math]. For example, if [math]\displaystyle{ \kappa }[/math] is supercompact and the generalized continuum hypothesis (GCH) holds below [math]\displaystyle{ \kappa }[/math] then it holds everywhere because a bijection between the powerset of [math]\displaystyle{ \nu }[/math] and a cardinal at least [math]\displaystyle{ \nu^{++} }[/math] would be a witness of limited rank for the failure of GCH at [math]\displaystyle{ \nu }[/math] so it would also have to exist below [math]\displaystyle{ \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]\displaystyle{ \kappa }[/math] such that for every structure [math]\displaystyle{ (M,R_1,\ldots,R_n) }[/math] with cardinality of the domain [math]\displaystyle{ \vert M\vert\geq\kappa }[/math], and for every [math]\displaystyle{ \Pi_1^1 }[/math] sentence [math]\displaystyle{ \phi }[/math] such that [math]\displaystyle{ (M,R_1,\ldots,R_n)\vDash\phi }[/math], there exists a substructure [math]\displaystyle{ (M',R_1\vert M,\ldots,R_n\vert M) }[/math] with smaller domain (i.e. [math]\displaystyle{ \vert M'\vert\lt \vert M\vert }[/math]) that satisfies [math]\displaystyle{ \phi }[/math].

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

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.