Diamondsuit
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In mathematics, and particularly in axiomatic set theory, 
(diamondsuit or diamond) is a certain family of combinatorial principles.
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[edit] Definition
For a given cardinal number κ and a stationary set 
, is the statement that there is a sequence 
such that
- each
- for every
is stationary in κ
When S = κ, is written 
, and 
is written 
[edit] Properties and use
It can be shown that ◊ ⇒ CH; also, ♣ + CH ⇒ ◊, but there also exist models of ♣ + ¬ CH, so ◊ and ♣ are not equivalent (rather, ♣ is weaker than ◊).
Charles Akemann and Nik Weaver used ◊ to construct a C*-algebra serving as a counterexample to Naimark's problem.
For all cardinals κ and stationary subsets 
, 
holds in the constructible universe.
[edit] References
- Charles Akemann, Nik Weaver, Consistency of a counterexample to Naimark's problem, online
