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| In the [[mathematics]] of [[transfinite number]]s, an '''ineffable cardinal''' is a certain kind of [[large cardinal]] number, introduced by {{harvtxt|Jensen|Kunen|1969}}.
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| 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 cardinal <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>.
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| A [[cardinal number]] <math>\kappa</math> is called '''ineffable''' if for every binary-valued function <math>f : \mathcal{P}_{=2}(\kappa) \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.
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| More generally, <math>\kappa</math> is called '''<math>n</math>-ineffable''' (for a positive integer <math>n</math>) if for every <math>f : \mathcal{P}_{=n}(\kappa) \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.
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| 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>.
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| Totally ineffable cardinals are of greater consistency strength than [[subtle cardinal]]s and of lesser consistency strength than [[remarkable cardinal]]s. A list of large cardinal axioms by consistency strength is available [[List_of_large_cardinal_properties | here]].
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| ==References==
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| *{{citation|doi=10.1016/S0168-0072(00)00019-1|first=Harvey|last=Friedman|authorlink=Harvey Friedman|title=Subtle cardinals and linear orderings|journal=Annals of Pure and Applied Logic|year=2001|volume=107|issue=1–3|pages=1–34}}.
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| *{{citation|title=Some Combinatorial Properties of L and V |url=http://www.mathematik.hu-berlin.de/~raesch/org/jensen.html|first=R. B. |last=Jensen|first2=K.|last2=Kunen|publisher=Unpublished manuscript|year=1969}}
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| [[Category:Large cardinals]]
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| {{settheory-stub}}
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