# Ineffable cardinal

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 ${\displaystyle \kappa }$ is called almost ineffable if for every ${\displaystyle f:\kappa \to {\mathcal {P}}(\kappa )}$ (where ${\displaystyle {\mathcal {P}}(\kappa )}$ is the powerset of ${\displaystyle \kappa }$) with the property that ${\displaystyle f(\delta )}$ is a subset of ${\displaystyle \delta }$ for all ordinals ${\displaystyle \delta <\kappa }$, there is a subset ${\displaystyle S}$ of ${\displaystyle \kappa }$ having cardinal ${\displaystyle \kappa }$ and homogeneous for ${\displaystyle f}$, in the sense that for any ${\displaystyle \delta _{1}<\delta _{2}}$ in ${\displaystyle S}$, ${\displaystyle f(\delta _{1})=f(\delta _{2})\cap \delta _{1}}$.

A cardinal number ${\displaystyle \kappa }$ is called ineffable if for every binary-valued function ${\displaystyle f:[\kappa ]^{2}\to \{0,1\}}$, there is a stationary subset of ${\displaystyle \kappa }$ on which ${\displaystyle f}$ is homogeneous: that is, either ${\displaystyle f}$ maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one.

More generally, ${\displaystyle \kappa }$ is called ${\displaystyle n}$-ineffable (for a positive integer ${\displaystyle n}$) if for every ${\displaystyle f:[\kappa ]^{n}\to \{0,1\}}$ there is a stationary subset of ${\displaystyle \kappa }$ on which ${\displaystyle f}$ is ${\displaystyle n}$-homogeneous (takes the same value for all unordered ${\displaystyle n}$-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 ${\displaystyle n}$-ineffable for every ${\displaystyle 2\leq n<\aleph _{0}}$. If ${\displaystyle \kappa }$ is ${\displaystyle (n+1)}$-ineffable, then the set of ${\displaystyle n}$-ineffable cardinals below ${\displaystyle \kappa }$ is a stationary subset of ${\displaystyle \kappa }$.

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.