Inductive limits of ideals

Abstract

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal $\mathcal{I}$ ($\text{rk}(\mathcal{I})$) as minimal ordinal $\alpha<\omega_{1}$ such that there is $\mathcal{S}\in\bf{\Sigma^0_{1+\alpha}}$ with $\mathcal{I}\subseteq \mathcal{S}$ and $\mathcal{I}^\star\cap \mathcal{S}=\emptyset$, where $\mathcal{I}^\star$ is the filter dual to the ideal $\mathcal{I}$ (actually, the authors use the dual notion of filters instead of ideals). Moreover, they introduced ideals $\text{Fin}_\alpha$, for all $\alpha<\omega_1$, and conjectured that $\text{rk}(\mathcal{I})\geq\alpha$ if and only if $\mathcal{I}$ contains an isomorphic copy of $\text{Fin}_\alpha$ ($\text{Fin}_\alpha\sqsubseteq\mathcal{I}$). To define $\text{Fin}_\alpha$ in the case of limit ordinals $0<\alpha<\omega_1$, G. Debs and J. Saint Raymond introduced inductive limits of ideals. We show that the above conjecture is false in the case of $\alpha=\omega$ by constructing an ideal $\text{Fin}'_\omega$ of rank $\omega$ such that $\text{Fin}_\omega\not\sqsubseteq\text{Fin}'_\omega$. However, we show that $\text{Fin}'_\omega\sqsubseteq\mathcal{I}$ is equivalent to $\forall_{n\in\omega}\text{Fin}_n\sqsubseteq\mathcal{I}$. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.