Ineffable cardinal
In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by (Jensen Kunen). In the following definitions, [math]\displaystyle{ \kappa }[/math] will always be a regular uncountable cardinal number.
A cardinal number [math]\displaystyle{ \kappa }[/math] is called almost ineffable if for every [math]\displaystyle{ f: \kappa \to \mathcal{P}(\kappa) }[/math] (where [math]\displaystyle{ \mathcal{P}(\kappa) }[/math] is the powerset of [math]\displaystyle{ \kappa }[/math]) with the property that [math]\displaystyle{ f(\delta) }[/math] is a subset of [math]\displaystyle{ \delta }[/math] for all ordinals [math]\displaystyle{ \delta \lt \kappa }[/math], there is a subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ \kappa }[/math] having cardinality [math]\displaystyle{ \kappa }[/math] and homogeneous for [math]\displaystyle{ f }[/math], in the sense that for any [math]\displaystyle{ \delta_1 \lt \delta_2 }[/math] in [math]\displaystyle{ S }[/math], [math]\displaystyle{ f(\delta_1) = f(\delta_2) \cap \delta_1 }[/math].
A cardinal number [math]\displaystyle{ \kappa }[/math] is called ineffable if for every binary-valued function [math]\displaystyle{ f : [\kappa]^2\to \{0,1\} }[/math], there is a stationary subset of [math]\displaystyle{ \kappa }[/math] on which [math]\displaystyle{ f }[/math] is homogeneous: that is, either [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ \kappa }[/math] is ineffable if for every sequence ⟨Aα : α ∈ κ⟩ such that each Aα ⊆ α, there is A ⊆ κ such that {α ∈ κ : A ∩ α = Aα} is stationary in κ.
Another equivalent formulation is that a regular uncountable cardinal [math]\displaystyle{ \kappa }[/math] is ineffable if for every set [math]\displaystyle{ S }[/math] of cardinality [math]\displaystyle{ \kappa }[/math] of subsets of [math]\displaystyle{ \kappa }[/math], there is a normal (i.e. closed under diagonal intersection) non-trivial [math]\displaystyle{ \kappa }[/math]-complete filter [math]\displaystyle{ \mathcal F }[/math] on [math]\displaystyle{ \kappa }[/math] deciding [math]\displaystyle{ S }[/math]: that is, for any [math]\displaystyle{ X\in S }[/math], either [math]\displaystyle{ X\in\mathcal F }[/math] or [math]\displaystyle{ \kappa\setminus X\in\mathcal F }[/math].[1] This is similar to a characterization of weakly compact cardinals.
More generally, [math]\displaystyle{ \kappa }[/math] is called [math]\displaystyle{ n }[/math]-ineffable (for a positive integer [math]\displaystyle{ n }[/math]) if for every [math]\displaystyle{ f : [\kappa]^n\to \{0,1\} }[/math] there is a stationary subset of [math]\displaystyle{ \kappa }[/math] on which [math]\displaystyle{ f }[/math] is [math]\displaystyle{ n }[/math]-homogeneous (takes the same value for all unordered [math]\displaystyle{ n }[/math]-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 [math]\displaystyle{ n }[/math]-ineffable for every [math]\displaystyle{ 2 \leq n \lt \aleph_0 }[/math]. If [math]\displaystyle{ \kappa }[/math] is [math]\displaystyle{ (n+1) }[/math]-ineffable, then the set of [math]\displaystyle{ n }[/math]-ineffable cardinals below [math]\displaystyle{ \kappa }[/math] is a stationary subset of [math]\displaystyle{ \kappa }[/math].
Every [math]\displaystyle{ n }[/math]-ineffable cardinal is [math]\displaystyle{ n }[/math]-almost ineffable (with set of [math]\displaystyle{ n }[/math]-almost ineffable below it stationary), and every [math]\displaystyle{ n }[/math]-almost ineffable is [math]\displaystyle{ n }[/math]-subtle (with set of [math]\displaystyle{ n }[/math]-subtle below it stationary). The least [math]\displaystyle{ n }[/math]-subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least [math]\displaystyle{ n }[/math]-almost ineffable is [math]\displaystyle{ \Pi^1_2 }[/math]-describable), but [math]\displaystyle{ (n-1) }[/math]-ineffable cardinals are stationary below every [math]\displaystyle{ n }[/math]-subtle cardinal.
A cardinal κ is completely ineffable if there is a non-empty [math]\displaystyle{ R \subseteq \mathcal{P}(\kappa) }[/math] such that
- every [math]\displaystyle{ A \in R }[/math] is stationary
- for every [math]\displaystyle{ A \in R }[/math] and [math]\displaystyle{ f : [\kappa]^2\to \{0,1\} }[/math], there is [math]\displaystyle{ B \subseteq A }[/math] homogeneous for f with [math]\displaystyle{ B \in R }[/math].
Using any finite [math]\displaystyle{ n }[/math] > 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 [math]\displaystyle{ \Pi^1_n }[/math]-indescribable for every n, but the property of being completely ineffable is [math]\displaystyle{ \Delta^2_1 }[/math].
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.
See also
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, http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html
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