Diamondsuit

In mathematics, and particularly in axiomatic set theory, ◊_S (diamondsuit or diamond) is a certain family of combinatorial principles.

Definition

For a given cardinal number κ and a stationary set S ⊆ κ, ◊_S is the statement that there is a sequence $\left\langle A_\delta: \delta \in S\right\rangle$ such that

every A_δ is a subset of δ
for every B ⊆ κ, the set $\left\{\alpha\in S: B\cap\alpha = A_\alpha\right\}$ is stationary

$\diamondsuit_{\omega_1}$ is usually written as just ◊.

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.

References

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

Diamondsuit

Definition

Properties and use

References

See also: Diamondsuit, Axiomatic set theory, C* algebra, Cardinal number, Clubsuit, Continuum hypothesis, Mathematics, Naimark's problem, Stationary set, Subset