Ineffable cardinal

In mathematics, an ineffable cardinal is a certain kind of large cardinal number.

Formally, a cardinal number κ is ineffable iff for every f: κ ² → {0, 1), there is a stationary subset of κ that is homogeneous for f.

κ is n-ineffable (for a positive integer n) iff for every f: κ ⁿ → {0, 1), there is a stationary subset of κ that is homogeneous for f.

A totally ineffable cardinal is a cardinal that is n-ineffable for every n. If κ is n+1-ineffable, then the set of n-ineffable cardinals below κ is a stationary subset of κ.