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

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

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

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.

References

• Friedman, Harvey (2001), "Subtle cardinals and linear orderings", Annals of Pure and Applied Logic, 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1.
• Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript