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