In mathematics, and particularly in axiomatic set theory, <math>\Diamond_\kappa (S)</math> (diamondsuit or diamond) is a certain family of combinatorial principles.
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Definition
For a given cardinal number <math>\kappa</math> and a stationary set <math> S\subseteq\kappa </math>, the statement <math> \Diamond_\kappa (S) </math> is the statement that there is a sequence <math>\langle A_\alpha: \alpha \in S \rangle </math> such that
- each <math> A_\alpha \subseteq \alpha </math>
- for every <math> A \subseteq \kappa, \{\alpha \in S: A \cap \alpha = A_\alpha\} </math> is stationary in <math>\kappa</math>
When <math> S = \kappa</math>, <math>\Diamond_\kappa (S) </math> is written <math> \Diamond_\kappa </math>, and <math> \Diamond_{\omega_1} </math> is written <math> \Diamond </math>
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 <math> \kappa </math> and stationary subsets <math> S \subseteq \kappa^+ </math>, <math> \Diamond_{\kappa^+} (S) </math> holds in the constructible universe.
References
- Charles Akemann, Nik Weaver, Consistency of a counterexample to Naimark's problem, online
See also
