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In [[mathematics]], and particularly in [[axiomatic set theory]], '''♣<sub>''S''</sub>''' ('''clubsuit''') is a family of [[Combinatorics|combinatorial principle]]s that are weaker version of the corresponding [[diamondsuit|◊<sub>''S''</sub>]]; it was introduced in 1975 <!--by A. Ostaszewski, but no article exists for him-->.
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== Definition ==
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For a given [[cardinal number]] <math>\kappa</math> and a [[stationary set]] <math>S \subseteq \kappa</math>, <math>\clubsuit_{S}</math> is the statement that there is a [[sequence]] <math>\left\langle A_\delta: \delta \in S\right\rangle</math> such that
 
* every ''A''<sub>''δ''</sub> is a cofinal [[subset]] of ''δ''
* for every [[Ordinal_number#Closed_unbounded_sets_and_classes|unbounded subset]] <math> A \subseteq \kappa</math>, there is a <math>\delta</math> so that <math>A_{\delta} \subseteq A</math>
<math>\clubsuit_{\omega_1}</math> is usually written as just <math>\clubsuit</math>.
 
== ♣ and ◊ ==
 
It is clear that ◊ ⇒ ♣, and it was shown in 1975 <!-- again by A. Ostaszewski--> that ♣ + [[continuum hypothesis|CH]] ⇒ ◊; however, [[Saharon Shelah]] gave a proof in 1980 that there exists a model of ♣ in which CH does not hold, so ♣ and ◊ are not equivalent (since ◊ ⇒ CH).
 
== References ==
 
* A. J. Ostaszewski, ''On countably compact perfectly [[normal space]]s'', Journal of [[London Mathematical Society]], 1975 (2) 14, pp. 505-516.
* S. Shelah, ''Whitehead groups may not be free, even assuming CH, II'', Israel Journal of Mathematics, 1980 (35) pp. 257-285.
 
== See also ==
*[[Club set]]
 
[[Category:Set theory]]

Latest revision as of 19:26, 13 May 2014

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