Supercompact cardinal

In set theory, a supercompact cardinal a type of large cardinal. They display a variety of reflection properties.

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

:$\{\; \}^lambda\; Msubseteq\; M$

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 &kappa; is supercompact if for every "A" such that |"A"| ≥ &kappa; there exists a normal measure on ["A"] ^{< &kappa;}.

["A"] ^{< &kappa;} is defined as follows:

:.

Properties

Supercompact cardinals have reflection properties. For example, if &kappa; is supercompact and the Generalized Continuum Hypothesis holds below &kappa; then it holds everywhere.

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

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