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Clubsuit

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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.

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

References

  1. Ostaszewski, Adam J. (1975). "On countably compact perfectly normal spaces". Journal of the London Mathematical Society. 14 (3): 505–516. doi:10.1112/jlms/s2-14.3.505.
  2. Shelah, S. (1980). "Whitehead groups may not be free even assuming CH, II". Israel Journal of Mathematics. 35 (4): 257–285. doi:10.1007/BF02760652.
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