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Ineffable cardinal
Kind of large cardinal number From Wikipedia, the free encyclopedia
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In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following definitions, will always be a regular uncountable cardinal number.
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 cardinality 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. An equivalent formulation is that a cardinal is ineffable if for every sequence such that each , there is such that is stationary in κ.
Another equivalent formulation is that a regular uncountable cardinal is ineffable if for every set of cardinality of subsets of , there is a normal (i.e. closed under diagonal intersection) non-trivial -complete filter on deciding : that is, for any , either or .[1] This is similar to a characterization of weakly compact cardinals.
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. Ineffability is strictly weaker than 3-ineffability.[2]p. 399
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 -ineffable cardinal is -almost ineffable (with set of -almost ineffable below it stationary), and every -almost ineffable is -subtle (with set of -subtle below it stationary). The least -subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least -almost ineffable is -describable), but -ineffable cardinals are stationary below every -subtle cardinal.
A cardinal κ is completely ineffable if there is a non-empty such that
- every is stationary
- for every and , there is homogeneous for f with .
Using any finite > 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 in the section below.
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