Supercompact cardinal

In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. 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 < κ, in the following sense. < κ is defined as follows: An ultrafilter U over < κ is fine if it is κ-complete and { x ∈ [ A ] < κ | a ∈ x } ∈ U {displaystyle {xin ^{<kappa }|ain x}in U} , for every a ∈ A {displaystyle ain A} . A normal measure over < κ is a fine ultrafilter U over < κ with the additional property that every function f : [ A ] < κ → A {displaystyle f:^{<kappa } o A} such that { x ∈ [ A ] < κ | f ( x ) ∈ x } ∈ U {displaystyle {xin ^{<kappa }|f(x)in x}in U} is constant on a set in U {displaystyle U} . Here 'constant on a set in U' means that there is a ∈ A {displaystyle ain A} such that { x ∈ [ A ] < κ | f ( x ) = a } ∈ U {displaystyle {xin ^{<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 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.