Supercompact cardinal

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

Formal definition[edit]

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 A such that \{x \in [A]^{< \kappa}| f(x)= a\} \in U .

Properties[edit]

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[edit]

Strongly compact cardinal

References[edit]

  • 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 ed.). Springer. ISBN 3-540-00384-3.