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In mathematics, and particularly in axiomatic set theory, ♣_S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊_S; it was introduced in 1975 by Adam Ostaszewski.^[1]

Definition

For a given cardinal number $\kappa$ and a stationary set $S\subseteq \kappa$ , $\clubsuit _{S}$ is the statement that there is a sequence $\left\langle A_{\delta }:\delta \in S\right\rangle$ such that

every A_δ is a cofinal subset of δ
for every unbounded subset $A\subseteq \kappa$ , there is a $\delta$ so that $A_{\delta }\subseteq A$

$\clubsuit _{\omega _{1}}$ is usually written as just $\clubsuit$ .

♣ and ◊

It is clear that ◊ ⇒ ♣, and it was shown in 1975 that ♣ + 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).^[2]

References

^ Ostaszewski, Adam J. (1975). "On countably compact perfectly normal spaces". Journal of the London Mathematical Society. 14 (3): 505–516. doi:10.1112/jlms/s2-14.3.505.
^ Shelah, S. (1980). "Whitehead groups may not be free even assuming CH, II". Israel Journal of Mathematics. 35 (4): 257–285. doi:10.1007/BF02760652.

[1] Ostaszewski, Adam J. (1975). "On countably compact perfectly normal spaces". Journal of the London Mathematical Society. 14 (3): 505–516. doi:10.1112/jlms/s2-14.3.505.

[2] Shelah, S. (1980). "Whitehead groups may not be free even assuming CH, II". Israel Journal of Mathematics. 35 (4): 257–285. doi:10.1007/BF02760652.

[1]

[2]

@@ Line 1: / Line 1: @@
-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]].
+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
-== Definition ==
+| 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 [[unbounded subset]] <math> A \subseteq \kappa</math>, there is a <math>\delta</math> so that <math>A_{\delta} \subseteq A</math>
+* 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 ◊ ==
+==♣ 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==
-It is clear that ◊ ⇒ ♣, and [[A. J. Ostaszewski]] showed in 1975 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).
+*[[Club set]]
-== References ==
+==References==
+{{reflist}}
-* 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]]
+[[Category:Mathematical principles]]