Clubsuit

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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 $\kappa$ 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).

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.

Clubsuit

Definition

♣ and ◊

References

See also