In mathematics, and particularly in axiomatic set theory, ♣<sub>S</sub> (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊<sub>S</sub>; it was introduced in 1975 by Adam Ostaszewski.

Definition

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

  • every A<sub>δ</sub> is a cofinal subset of δ
  • for every unbounded subset <math> A \subseteq \kappa</math>, there is a <math>\delta</math> so that <math>A_{\delta} \subseteq A</math>

<math>\clubsuit_{\omega_1}</math> is usually written as just <math>\clubsuit</math>.

♣ and ◊

It is clear that ◊ ⇒ ♣, and it was shown in 1975 <!-- again by A. Ostaszewski--> 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

References