Puneet Varma (Editor)

Clubsuit

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In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are weaker version of the corresponding ◊S; it was introduced in 1975 .

Contents

Definition

For a given cardinal number κ and a stationary set S ⊆ κ , ♣ S is the statement that there is a sequence ⟨ A δ : δ ∈ S ⟩ such that

  • every Aδ is a cofinal subset of δ
  • for every unbounded subset A ⊆ κ , there is a δ so that A δ ⊆ A

    • ♣ ω 1 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 (since ◊ ⇒ CH).

    References

    Clubsuit Wikipedia
    (Text) CC BY-SA