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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 a weaker version of the corresponding [[diamondsuit|◊<sub>''S''</sub>]]; it was introduced in 1975 by Adam Ostaszewski.<ref>{{cite journal
| last1=Ostaszewski | first1=Adam J.
| title=On countably compact perfectly [[normal space]]s
| journal=[[Journal of the London Mathematical Society]]
| year=1975
| volume=14
| issue=3
| pages=505–516
| doi=10.1112/jlms/s2-14.3.505}}</ref>

==Definition==
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 [[Diamond principle|◊]] ⇒ ♣, 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).<ref>{{cite journal
| last1=Shelah | first1=S.
| title=Whitehead groups may not be free even assuming CH, II
| journal=[[Israel Journal of Mathematics]]
| year=1980
| volume=35
| issue=4
| pages=257–285
| doi=10.1007/BF02760652 | doi-access=free}}</ref>

==See also==
*[[Club set]]

==References==
{{reflist}}

[[Category:Set theory]]
[[Category:Mathematical principles]]