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).

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 .

Totally ineffable cardinals are of greater consistency strength than subtle cardinals and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.