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 by A. Ostaszewski.

Definition

For a given cardinal number κ and a stationary set Sκ, ♣S is the statement that there is a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle A_\delta: \delta \in S\right\rangle} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \clubsuit_{\omega_1}} is usually written as just ♣.

♣ and ◊

It is clear that ◊ ⇒ ♣, and A. J. Ostaszewski showed in 1975 that ♣ + CH ⇒ ◊; however, Saharon Shelah gave proof in 1980 that there exists a model of ♣ in which CH does not hold, so ♣ and ◊ are not equivalent (since ◊ ⇒ CH).

References

  • 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.