Editing Ineffable cardinal

{{short description|Kind of large cardinal number}}
In the [[mathematics]] of [[transfinite number]]s, an '''ineffable cardinal''' is a certain kind of [[large cardinal]] number, introduced by {{harvtxt|Jensen|Kunen|1969}}. In the following definitions, <math>\kappa</math> will always be a [[regular cardinal|regular]] [[uncountable set|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 (large cardinal property)|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 (large cardinal property)|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]] <math>\langle A_\alpha:\alpha\in\kappa\rangle</math> such that each <math>A_\alpha\subseteq\alpha</math>, 
there is <math>A\subseteq\kappa</math> such that <math>\{\alpha\in\kappa:A\cap\alpha=A_\alpha\}</math> is stationary in {{math|''κ''}}.

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]]) [[Filter (set theory)|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>.<ref>{{cite arXiv|eprint=1710.10043 |last1=Holy |first1=Peter |last2=Schlicht |first2=Philipp |title=A hierarchy of Ramsey-like cardinals |date=2017 |class=math.LO }}</ref> This is similar to a characterization of [[weakly compact cardinal]]s.

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 (large cardinal property)|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. Ineffability is strictly weaker than 3-ineffability.<ref>K. Kunen,. "Combinatorics". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref><sup>p. 399</sup>

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 cardinal|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>-[[Indescribable cardinal|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<br/>
- every <math>A \in R</math> is stationary<br/>
- 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>&nbsp;>&nbsp;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 cardinal|iterable cardinals]], which in turn is below [[Remarkable cardinal|remarkable cardinals]], which in turn is below [[Erdős cardinal|ω-Erdős]] cardinals. A list of large cardinal axioms by consistency strength is available in the section below.

==See also==
* [[List of large cardinal properties]]

==References==
*{{citation|doi=10.1016/S0168-0072(00)00019-1|first=Harvey|last=Friedman|authorlink=Harvey Friedman (mathematician)|title=Subtle cardinals and linear orderings|journal=Annals of Pure and Applied Logic|year=2001|volume=107|issue=1–3|pages=1–34|doi-access=free}}. 
*{{citation|title=Some Combinatorial Properties of L and V |url=http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html |first1=Ronald|last1=Jensen|authorlink=Ronald Jensen |first2=Kenneth|last2=Kunen|author2-link=Kenneth Kunen |publisher=Unpublished manuscript|year=1969}}
===Citations===
{{Reflist}}

[[Category:Large cardinals]]