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Ineffable cardinal

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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 ⟨Aα : α ∈ κ⟩ such that each Aα ⊆ α, there is Aκ such that {ακ : Aα = Aα} 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.

See also

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References

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  • Friedman, Harvey (2001), "Subtle cardinals and linear orderings", Annals of Pure and Applied Logic, 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1.
  • Jensen, Ronald; Kunen, Kenneth (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

Citations

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  1. ^ Holy, Peter; Schlicht, Philipp (2017). "A hierarchy of Ramsey-like cardinals". arXiv:1710.10043 [math.LO].
  2. ^ K. Kunen,. "Combinatorics". In Handbook of Mathematical Logic, Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)