Clubsuit Explained

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.^[1]

Definition

\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

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).^[2]

See also

• Club set

Notes and References

1. Ostaszewski . Adam J.. On countably compact perfectly normal spaces. Journal of the London Mathematical Society. 1975. 14. 505–516. 10.1112/jlms/s2-14.3.505.
2. Shelah . S.. Whitehead groups may not be free even assuming CH, II. Israel Journal of Mathematics. 1980. 35. 257–285. 10.1007/BF02760652 . free.