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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-->. | |||
== 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 ◊ ⇒ ♣, 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]] | |||
Revision as of 10:11, 13 March 2013
In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are weaker version of the corresponding ◊S; it was introduced in 1975 .
Definition
For a given cardinal number and a stationary set , is the statement that there is a sequence such that
- every Aδ is a cofinal subset of δ
- for every unbounded subset , there is a so that
is usually written as just .
♣ 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).
References
- A. J. Ostaszewski, On countably compact perfectly normal spaces, 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.