In set theory, a supercompact cardinal is 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
\( {}^{\lambda }M\subseteq M\,. \)
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 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:
\( [A]^{{<\kappa }}:=\{x\subseteq A||x|<\kappa \}\,. \)
An ultrafilter U over [A]< κ is fine if it is κ-complete and \( \{x\in [A]^{{<\kappa }}|a\in x\}\in U \) , for every \( a\in A \). A normal measure over [A]< κ is a fine ultrafilter U over [A]< κ with the additional property that every function \( f:[A]^{{<\kappa }}\to A \) such that \( \{x\in [A]^{{<\kappa }}|f(x)\in x\}\in U \) is constant on a set in U. Here "constant on a set in U" means that there is \( a\in \) such that \( \{x\in [A]^{{<\kappa }}|f(x)=a\}\in U \).
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.
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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