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 .

Definition[edit]

For a given cardinal number \kappa and a stationary set S \subseteq \kappa, \clubsuit_{S} is the statement that there is a sequence \left\langle A_\delta: \delta \in S\right\rangle such that

\clubsuit_{\omega_1} is usually written as just \clubsuit.

♣ and ◊[edit]

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[edit]

  • A. J. Ostaszewski, On countably compact perfectly normal spaces, Journal of London Mathematical Society, 1975 (2) 14, pp. 505-516.
  • S. Shelah, Whitehead groups may not be free, even assuming CH, II, Israel Journal of Mathematics, 1980 (35) pp. 257-285.

See also[edit]