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# Clubsuit

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 by A. Ostaszewski .

## Definition

For a given cardinal number κ and a stationary set Sκ, ♣S is the statement that there is a sequence $\left\langle A_\delta: \delta \in S\right\rangle$ such that

• every Aδ is a subset of δ
• for every unbounded subset Aκ, there is a δ so that AδA

$\clubsuit_{\omega_1}$ is usually written as just ♣.

## ♣ and ◊

It is clear that ◊ ⇒ ♣, and A. J. Ostaszewski showed in 1975 that ♣ + CH ⇒ ◊; however, Saharon Shelah gave 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.
Last updated: 08-07-2005 16:07:14
03-10-2013 05:06:04
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