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 S ⊆ κ, ♣_S is the statement that there is a sequence <math>\left\langle A_\delta: \delta \in S\right\rangle<math> such that

• every A_δ is a subset of δ
• for every unbounded subset A ⊆ κ, there is a δ so that A_δ ⊆ A

<math>\clubsuit_{\omega_1}<math> 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.