Diamond principle (set theory) - Definition and Overview

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 <math>\left\langle A_\delta: \delta \in S\right\rangle<math> such that

• every A_δ a subset of δ
• for every A ⊆ κ, the set <math>\left\{\alpha\in S: A\cap\alpha = A_\alpha\right\}<math> is stationary

<math>\diamondsuit_{\omega_1}<math> 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 (http://arxiv.org/abs/math.OA/0312135)