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Diamondsuit
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DEFINITION


For a given Cardinal Number \kappa and a Stationary Set S\subseteq\kappa , \Diamond_\kappa (S) is the statement that there is a Sequence \langle A_\alpha: \alpha \in S angle such that

  • each A_\alpha \subseteq \alpha

  • for every A \subseteq \kappa, \{\alpha \in S: A \cap \alpha = A_\alpha\} is stationary in \kappa


When \Diamond_\kappa (S) is written \Diamond_\kappa , and \Diamond_{\omega_1} is written \Diamond


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 ◊).


For all cardinals \kappa and stationary subsets S \subseteq \kappa^+ , \Diamond_{\kappa^+} (S) holds in the Constructible Universe .


REFERENCES


  • Charles Akemann, Nik Weaver, ''Consistency of a counterexample to Naimark's problem'', online



SEE ALSO