Elasticity (physics): Difference between revisions

Revision as of 02:26, 30 January 2014

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Template:Harvtxt.

A cardinal number $κ$ is called almost ineffable if for every $f : κ \to 𝒫 (κ)$ (where $𝒫 (κ)$ is the powerset of $κ$ ) with the property that $f (δ)$ is a subset of $δ$ for all ordinals $δ < κ$ , there is a subset $S$ of $κ$ having cardinal $κ$ and homogeneous for $f$ , in the sense that for any $δ_{1} < δ_{2}$ in $S$ , $f (δ_{1}) = f (δ_{2}) \cap δ_{1}$ .

A cardinal number $κ$ is called ineffable if for every binary-valued function $f : 𝒫_{= 2} (κ) \to {0, 1}$ , there is a stationary subset of $κ$ 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.

More generally, $κ$ is called $n$ -ineffable (for a positive integer $n$ ) if for every $f : 𝒫_{= n} (κ) \to {0, 1}$ there is a stationary subset of $κ$ 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.

A totally ineffable cardinal is a cardinal that is $n$ -ineffable for every $2 \leq n < ℵ_{0}$ . If $κ$ is $(n + 1)$ -ineffable, then the set of $n$ -ineffable cardinals below $κ$ is a stationary subset of $κ$ .

Totally ineffable cardinals are of greater consistency strength than subtle cardinals and of lesser consistency strength than remarkable cardinals. A list of large cardinal axioms by consistency strength is available here.

References

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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}}.
+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>.
+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.
+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.
+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>.
+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]].
+==References==
+*{{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}}.
+*{{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}}
+[[Category:Large cardinals]]
+{{settheory-stub}}

Elasticity (physics): Difference between revisions

Revision as of 02:26, 30 January 2014

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