“Ineffable cardinal”的意思、由来-开放百科全书

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Ineffable cardinal

释义

References

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by {{harvtxt|Jensen|Kunen|1969}}.

A cardinal number is called almost ineffable if for every (where is the powerset of ) with the property that is a subset of for all ordinals , there is a subset of having cardinal and homogeneous for , in the sense that for any in , .

A cardinal number is called ineffable if for every binary-valued function , there is a stationary subset of on which is homogeneous: that is, either maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.

More generally, is called -ineffable (for a positive integer ) if for every there is a stationary subset of on which is -homogeneous (takes the same value for all unordered -tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.

A totally ineffable cardinal is a cardinal that is -ineffable for every . If is -ineffable, then the set of -ineffable cardinals below is a stationary subset of .

Every n-ineffable cardinal is n-almost ineffable (with set of n-almost ineffable below it stationary), and every n-almost ineffable is n-subtle (with set of n-subtle below it stationary). The least n-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least n-almost ineffable is -describable), but n-1-ineffable cardinals are stationary below every n-subtle cardinal.

A cardinal κ is completely ineffable iff there is a non-empty such that

- every

is stationary
- for every

and

, there is

homogeneous for f with

Using any finite n>1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater consistency strength). Completely ineffable cardinals are -indescribable for every n, but the property of being completely ineffable is .

The consistency strength of completely ineffable is below that of 1-iterable cardinals, which in turn is below remarkable cardinals, which in turn is below ω-Erdős cardinals. A list of large cardinal axioms by consistency strength is available here.

References

{{citation|doi=10.1016/S0168-0072(00)00019-1|first=Harvey|last=Friedman|authorlink=Harvey Friedman|title=Subtle cardinals and linear orderings|journal=Annals of Pure and Applied Logic|year=2001|volume=107|issue=1–3|pages=1–34}}.
{{citation|title=Some Combinatorial Properties of L and V |url=http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html |first=Ronald|last=Jensen|authorlink=Ronald Jensen |first2=Kenneth|last2=Kunen|author2-link=Kenneth Kunen |publisher=Unpublished manuscript|year=1969}}

1 : Large cardinals

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