Ineffable cardinal

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by. 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 $⟨A_{α} : α ∈ κ⟩$ such that each $A_{α} ⊆ α$, there is $A ⊆ κ$ such that ${α ∈ κ : A ∩ α = A_{α} }$ 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 $$\Pi^1_2$$-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 $$\Pi^1_n$$-indescribable for every n, but the property of being completely ineffable is $$\Delta^2_1$$.

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.