Supercompact cardinal

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 [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 ${\displaystyle \{x\in [A]^{<\kappa }|a\in x\}\in U}$, for every ${\displaystyle a\in A}$. A normal measure over [A]^{< κ} is a fine ultrafilter U over [A]^{< κ} with the additional property that every function ${\displaystyle f:[A]^{<\kappa }\to A}$ such that ${\displaystyle \{x\in [A]^{<\kappa }|f(x)\in x\}\in U}$ is constant on a set in ${\displaystyle U}$. Here "constant on a set in U" means that there is ${\displaystyle a\in A}$ such that ${\displaystyle \{x\in [A]^{<\kappa }|f(x)=a\}\in U}$.

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