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.
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 iff 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.