Supercompact cardinal

Formal definition

If λ is any ordinal, κ is λ-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ, j(κ)>λ and

That is, M contains all of its λ-sequences. Then κ is supercompact means that it is λ-supercompact for all ordinals λ.

Alternatively, an uncountable cardinal κ is supercompact if for every A such that |A| ≥ κ there exists a normal measure over [A]^{< κ}, in the following sense.

[A]^{< κ} is defined as follows:

An ultrafilter U over [A]^{< κ} is fine if it is κ-complete and { x ∈ [ A ] < κ | a ∈ x } ∈ U , for every a ∈ A . A normal measure over [A]^{< κ} is a fine ultrafilter U over [A]^{< κ} with the additional property that every function f : [ A ] < κ → A such that { x ∈ [ A ] < κ | f ( x ) ∈ x } ∈ U is constant on a set in U . Here "constant on a set in U" means that there is a ∈ A such that { x ∈ [ A ] < κ | f ( x ) = a } ∈ 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 κ, then a cardinal with that property exists below κ. For example, if κ is supercompact and the Generalized Continuum Hypothesis holds below κ then it holds everywhere because a bijection between the powerset of ν and a cardinal at least ν⁺⁺ would be a witness of limited rank for the failure of GCH at ν so it would also have to exist below κ.

Finding a canonical inner model for supercompact cardinals is one of the major problems of inner model theory.