Ineffable cardinal
In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions, formula_1 will always be a regular uncountable cardinal number. A cardinal number formula_1 is called almost ineffable if for every formula_3 (where formula_4 is the powerset of formula_1) with the property that formula_6 is a subset of formula_7 for all ordinals formula_8, there is a subset formula_9 of formula_1 having cardinality formula_1 and homogeneous for formula_12, in the sense that for any formula_13 in formula_9, formula_15. A cardinal number formula_1 is called ineffable if for  every binary-valued function formula_17, there is a stationary subset of formula_1 on which formula_12 is homogeneous: that is, either formula_12 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 formula_1 is ineffable if for every sequence such that each , there is such that is stationary in . More generally, formula_1 is called formula_23-ineffable (for a positive integer formula_23) if for every formula_25 there is a stationary subset of formula_1 on which formula_12 is formula_23-homogeneous (takes the same value for  all unordered formula_23-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 formula_23-ineffable for every formula_31. If formula_1 is formula_33-ineffable, then the set of formula_23-ineffable cardinals below formula_1 is a stationary subset of formula_1. 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 formula_37 -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 formula_38 such that - every formula_39 is stationary - for every formula_39 and formula_17, there is formula_42 homogeneous for "f" with formula_43. 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 formula_44-indescribable for every "n", but the property of being completely ineffable is formula_45. 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.