Information About

Clubsuit





DEFINITION


For a given Cardinal Number ''κ'' and a Stationary Set ''S'' ⊆ ''κ'', ♣''S'' is the statement that there is a Sequence \left\langle A_\delta: \delta \in S ight angle such that

  • every ''A''''δ'' is a Subset of ''δ''

  • for every Unbounded Subset ''A'' ⊆ ''κ'', there is a ''δ'' so that ''A''''δ'' ⊆ ''A''


\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 Space s'', 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.