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.
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).
- 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.
- Shelah, S. (1980). "Whitehead groups may not be free even assuming CH, II". Israel Journal of Mathematics. 35: 257–285. doi:10.1007/BF02760652.