Ineffable cardinal

Large cardinals

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . 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 A ⊆ κ such that {α ∈ κ : A ∩ α = Aα} is stationary in κ. 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 if 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. (Wikipedia).

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Ineffable cardinal

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