Club principle

In mathematics, and particularly in axiomatic set theory, the club principles ♣_S are 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 $\kappa$ and a stationary set $S\subseteq \kappa$, ${\mathrm{♣}}_S$ is the statement that there is a sequence $⟨A_{\delta }:\delta \in S⟩$ such that

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

${\mathrm{♣}}_{{\omega }_1}$ is usually written as just $\mathrm{♣}$.

♣ 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