“Supercompact cardinal”的意思、由来-开放百科全书

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Supercompact cardinal

释义

Formal definition
Properties
See also
References

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

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 , for every . A normal measure over [A]^{< κ} is a fine ultrafilter U over [A]^{< κ} with the additional property that every function such that is constant on a set in . Here "constant on a set in U" means that there is such that .

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.

References

{{cite book|author=Drake, F. R.|title=Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76)|publisher=Elsevier Science Ltd|year=1974|isbn=0-444-10535-2}}
{{cite book|author=Jech, Thomas|title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=3-540-44085-2|authorlink=Thomas Jech}}
{{cite book|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|edition=2nd|isbn=3-540-00384-3}}

1 : Large cardinals

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Formal definition

Properties

See also

References