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
*A*_{δ}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

**^**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.