In mathematics, and particularly in axiomatic set theory, the club principles ♣S are a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ostaszewski.
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).
See also Club set
References