Inductive limits of ideals
Adam Kwela
TL;DR
The paper investigates the Borel separation rank $rk(\mathcal{I})$ of analytic ideals and the conjectured equivalence with containing copies of $Fin_\alpha$. It develops the inductive-limit framework for defining $Fin_\alpha$ at limit ordinals and constructs a rank-$\omega$ counterexample $Fin'_\omega$ that does not embed $Fin_\omega$, while showing $Fin'_\omega$ embeds all finite-stage $Fin_n$, yielding $Fin'_\omega\leq\mathcal{I}$ iff $Fin_n\leq\mathcal{I}$ for all $n$. This leads to a proposed modification of the conjecture for limit ordinals, where $Fin'_\omega$ (and more generally $Fin'_\alpha$) replaces $Fin_\alpha$ as the canonical limit-object, aligning the rank with all finite stages and guiding potential generalizations to other limit ordinals. The results offer a nuanced view of how descriptive complexity and embeddings interact in the rank theory of analytic ideals, with implications for the structure of ideal pointwise limits and their descriptive complexity.
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 $α<ω_{1}$ such that there is $\mathcal{S}\in\bf{Σ^0_{1+α}}$ 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 $\text{Fin}_α$, for all $α<ω_1$, and conjectured that $\text{rk}(\mathcal{I})\geqα$ if and only if $\mathcal{I}$ contains an isomorphic copy of $\text{Fin}_α$ ($\text{Fin}_α\sqsubseteq\mathcal{I}$). To define $\text{Fin}_α$ in the case of limit ordinals $0<α<ω_1$, G. Debs and J. Saint Raymond introduced inductive limits of ideals. We show that the above conjecture is false in the case of $α=ω$ by constructing an ideal $\text{Fin}'_ω$ of rank $ω$ such that $\text{Fin}_ω\not\sqsubseteq\text{Fin}'_ω$. However, we show that $\text{Fin}'_ω\sqsubseteq\mathcal{I}$ is equivalent to $\forall_{n\inω}\text{Fin}_n\sqsubseteq\mathcal{I}$. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.