Diamondsuit
From Biocrawler, the free encyclopedia.
(Redirected from Diamond principle (set theory))
In mathematics, and particularly in axiomatic set theory, ◊S (diamondsuit or diamond) is a certain family of combinatorial principles.
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Definition
For a given cardinal number κ and a stationary set S ⊆ κ, ◊S is the statement that there is a sequence 
such that
- every Aδ is a subset of δ
- for every B ⊆ κ, the set
is stationary
is usually written as just ◊.
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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.
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References
- Charles Akemann, Nik Weaver, Consistency of a counterexample to Naimark's problem, online (http://arxiv.org/abs/math.OA/0312135)
