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 $⟨A α : α \in κ⟩$ such that each $A α \subseteq α$ , there is $A \subseteq κ$ such that ${α \in κ : A \cap α = A α}$ is stationary in $κ$ .

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.

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 iff 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 $\Pi^1_n$ -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 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, Ronald; Kunen, Kenneth (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

Ineffable cardinal

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References