Clubsuit

In mathematics, and particularly in axiomatic set theory, ♣_S 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 and a stationary set, is the statement that there is a sequence such that

every A_δ is a cofinal subset of δ
for every unbounded subset, there is a so that

is usually written as just.

♣ 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.