Diamond (mathematics)

In mathematics, and particularly in axiomatic set theory, $$Diamond_kappa (S) (diamond) is a certain family of combinatorial principles.

Definition

For a given cardinal number $$kappa and a stationary set $$Ssubseteqkappa , the statement $$Diamond_kappa (S) is the statement that there is a sequence $$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 $$S = kappa, $$Diamond_kappa (S) is written $$Diamond_kappa , and $$Diamond_{omega_1} is written $$Diamond

Properties and use

Diamond principles are classically used to build Suslin trees.

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 $$kappa and stationary subsets $$S subseteq kappa^+ , $$Diamond_{kappa^+} (S) holds in the constructible universe. Recently Shelah proved that for $$kappa>aleph_0, $$diamondsuit_{kappa^+} follows from $$2^kappa=kappa^+.

References

* Charles Akemann, Nik Weaver, "Consistency of a counterexample to Naimark's problem", [http://arxiv.org/abs/math.OA/0312135 online]

ee also

*statements true in L
*axiom of constructibility