$\mathrm{◇}$

Definition 1.

Let $S\mathrm{\subseteq }\kappa$ be a stationary set. Then the combinatorial principle ${\mathrm{◇}}_S$ holds if and only if there is a sequence ${\mathrm{⟨}{A}_{\alpha }\mathrm{⟩}}_{\alpha \mathrm{\in }S}$ such that each $A_{\alpha }\mathrm{\subseteq }\alpha$ and for any $A\mathrm{\subseteq }\kappa$, $\mathrm{\{}\alpha \mathrm{\in }S\mathrm{\mid }A\mathrm{\cap }\alpha \mathrm{=}{A}_{\alpha }\mathrm{\}}$ is stationary.

To get some sense of what this means, observe that for any , $\{\lambda \}\subseteq \kappa$, so the set of $A_{\alpha }=\{\lambda \}$ is stationary (in $\kappa$). More strongly, suppose $\kappa >\lambda$. Then any subset of $T\subset \lambda$ is bounded in $\kappa$ so $A_{\alpha }=T$ on a stationary set. Since $|S|=\kappa$, it follows that $2^{\lambda }\le \kappa$. Hence ${\mathrm{◇}}_{{\mathrm{\aleph }}_1}$, the most common form (often written as just $\mathrm{◇}$), implies CH.

C. Akemann and N. Weaver used $\mathrm{◇}$ to construct a $C^{*}$-algebra serving as a counterexample to Naimark’s problem.