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From Wikipedia, the free encyclopedia

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.[1]

Definition

For a given cardinal number and a stationary set , is the statement that there is a sequence such that

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

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).[2]

See also

References

  1. ^ Ostaszewski, Adam J. (1975). "On countably compact perfectly normal spaces". Journal of the London Mathematical Society. 14: 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: 257–285. doi:10.1007/BF02760652.
This page was last edited on 8 October 2021, at 09:56
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