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.
- 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- Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript
|This set theory-related article is a stub. You can help Wikipedia by expanding it.|