# Ineffable cardinal

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following definitions, $\kappa$ will always be a regular uncountable cardinal number.
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 cardinality $\kappa$ 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 $f:[\kappa ]^{2}\to \{0,1\}$, 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. An equivalent formulation is that a cardinal $\kappa$ is ineffable if for every sequence $\langle A_{\alpha }:\alpha \in \kappa \rangle$ such that each $A_{\alpha }\subseteq \alpha$, there is $A\subseteq \kappa$ such that $\{\alpha \in \kappa :A\cap \alpha =A_{\alpha }\}$ is stationary in κ.
Another equivalent formulation is that a regular uncountable cardinal $\kappa$ is ineffable if for every set $S$ of cardinality $\kappa$ of subsets of $\kappa$, there is a normal (i.e. closed under diagonal intersection) non-trivial $\kappa$-complete filter ${\mathcal {F}}$ on $\kappa$ deciding $S$: that is, for any $X\in S$, either $X\in {\mathcal {F}}$ or $\kappa \setminus X\in {\mathcal {F}}$. This is similar to a characterization of weakly compact cardinals.
More generally, $\kappa$ is called $n$-ineffable (for a positive integer $n$) if for every $f:[\kappa ]^{n}\to \{0,1\}$ 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. Ineffability is strictly weaker than 3-ineffability.$^{p.399}$
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 $\kappa$.
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 $\Pi _{2}^{1}$-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 $R\subseteq {\mathcal {P}}(\kappa )$ such that - every $A\in R$ is stationary - for every $A\in R$ and $f:[\kappa ]^{2}\to \{0,1\}$, there is $B\subseteq A$ homogeneous for f with $B\in R$.
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 $\Pi _{n}^{1}$-indescribable for every n, but the property of being completely ineffable is $\Delta _{1}^{2}$.
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 in the section below.