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 by Adam Ostaszewski.

Definition

For a given cardinal number [math]\displaystyle{ \kappa }[/math] and a stationary set [math]\displaystyle{ S \subseteq \kappa }[/math], [math]\displaystyle{ \clubsuit_{S} }[/math] is the statement that there is a sequence [math]\displaystyle{ \left\langle A_\delta: \delta \in S\right\rangle }[/math] such that

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

[math]\displaystyle{ \clubsuit_{\omega_1} }[/math] is usually written as just [math]\displaystyle{ \clubsuit }[/math].

♣ 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).

See also

• Club set