Clubsuit

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.

Definition

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

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

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

♣ 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.

See also

• Club set