Clubsuit
Appearance
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
[edit]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 ◊
[edit]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
[edit]References
[edit]- ^ Ostaszewski, Adam J. (1975). "On countably compact perfectly normal spaces". Journal of the London Mathematical Society. 14 (3): 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 (4): 257–285. doi:10.1007/BF02760652.