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, ${\displaystyle \kappa }$ will always be a regular uncountable cardinal number.

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 cardinality ${\displaystyle \kappa }$ 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. An equivalent formulation is that a cardinal ${\displaystyle \kappa }$ is ineffable if for every sequence ⟨A_α : α ∈ κ⟩ such that each A_α ⊆ α, there is A ⊆ κ such that {α ∈ κ : A ∩ α = A_α} is stationary in κ.

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 ${\displaystyle \kappa }$.

Every ${\displaystyle n}$-ineffable cardinal is ${\displaystyle n}$-almost ineffable (with set of ${\displaystyle n}$-almost ineffable below it stationary), and every ${\displaystyle n}$-almost ineffable is ${\displaystyle n}$-subtle (with set of ${\displaystyle n}$-subtle below it stationary). The least ${\displaystyle n}$-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least ${\displaystyle n}$-almost ineffable is ${\displaystyle \Pi _{2}^{1}}$-describable), but ${\displaystyle (n-1)}$-ineffable cardinals are stationary below every ${\displaystyle n}$-subtle cardinal.

A cardinal κ is completely ineffable if there is a non-empty ${\displaystyle R\subseteq {\mathcal {P}}(\kappa )}$ such that
- every ${\displaystyle A\in R}$ is stationary
- for every ${\displaystyle A\in R}$ and ${\displaystyle f:[\kappa ]^{2}\to \{0,1\}}$, there is ${\displaystyle B\subseteq A}$ homogeneous for f with ${\displaystyle B\in R}$.

Using any finite ${\displaystyle n}$ > 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 ${\displaystyle \Pi _{n}^{1}}$-indescribable for every n, but the property of being completely ineffable is ${\displaystyle \Delta _{1}^{2}}$.

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.