◇ (original) (raw)

◇

Definition 1.

To get some sense of what this means, observe that for any λ<κ, {λ}⊆κ, so the set of Aα={λ} is stationary (in κ). More strongly, suppose κ>λ. Then any subset of T⊂λ is bounded in κ so Aα=T on a stationary set. Since |S|=κ, it follows that 2λ≤κ. Hence ◇ℵ1, the most common form (often written as just ◇), implies CH.

C. Akemann and N. Weaver used ◇ to construct a C*-algebra serving as a counterexample to Naimark’s problem.

References

• 1 Akemann, C., and N. Weaver, Consistency of a counterexample to Naimark’s problem. Preprint available on the arXiv at http://arxiv.org/abs/math.OA/0312135http://arxiv.org/abs/math.OA/0312135.