Clubsuit
Encyclopedia
In mathematics
, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principle
s that are weaker version of the corresponding ◊S
; it was introduced in 1975 by A. Ostaszewski.
and a stationary set
, 
is the statement that there is a sequence
such that
is usually written as just 
.
⇒ ◊; 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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principle
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
s that are weaker version of the corresponding ◊S
Diamondsuit
In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by that holds in the constructible universe and that implies the continuum hypothesis...
; it was introduced in 1975 by A. Ostaszewski.
Definition
For a given cardinal numberCardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
and a stationary setStationary set
In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:-Classical notion:If \kappa \, is a cardinal of uncountable cofinality, S \subseteq \kappa \,, and S \, intersects every club set in \kappa \,, then S \, is called a stationary set....
, is the statement that there is a sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
such that- every Aδ is a cofinal subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of δ - for every unbounded subset
, there is a 
so that 
is usually written as just .♣ and ◊
It is clear that ◊ ⇒ ♣, and A. J. Ostaszewski showed in 1975 that ♣ + CHContinuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
⇒ ◊; however, Saharon Shelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
gave a proof in 1980 that there exists a model of ♣ in which CH does not hold, so ♣ and ◊ are not equivalent (since ◊ ⇒ CH).