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 κ {\displaystyle \kappa } and a stationary set S ⊆ κ {\displaystyle S\subseteq \kappa } , ♣ S {\displaystyle \clubsuit _{S}} is the statement that there is a sequence ⟨ A δ : δ ∈ S ⟩ {\displaystyle \left\langle A_{\delta }:\delta \in S\right\rangle } such that
every Aδ is a cofinal subset of δ for every unbounded subset A ⊆ κ {\displaystyle A\subseteq \kappa } , there is a δ {\displaystyle \delta } so that A δ ⊆ A {\displaystyle A_{\delta }\subseteq A}
♣ ω 1 {\displaystyle \clubsuit _{\omega _{1}}} is usually written as just ♣ {\displaystyle \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).
See also Club set
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