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 ${\displaystyle S}$, ${\displaystyle f(\delta _{1})=f(\delta _{2})\cap \delta _{1}}$.

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