diamondsuit

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

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

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