On a conjecture of Debs and Saint Raymond
Adam Kwela
TL;DR
The paper addresses whether analytic ideals of a given Borel separation rank must embed Fin_alpha; it provides a negative answer for alpha = 3 by constructing a Sigma0-6 CEI on omega^4 with rk(CEI) > 2 and Fin3 not below CEI in the Katětov order. The construction uses a Fin3-based Finite product and a diagonalization argument to rule out Sigma0_3 separators, establishing the rank bound. This result clarifies the relation between Borel complexity and embedding properties of finite-stage Fin hierarchies within ideals and informs the study of ideal pointwise limits in descriptive set theory.
Abstract
Borel separation rank of an analytic ideal $\mathcal{I}$ on $ω$ is the 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 $\mathcal{I}$. Answering in negative a question of G. Debs and J. Saint Raymond [Fund. Math. 204 (2009), no. 3], we construct a Borel ideal of rank $>2$ which does not contain an isomorphic copy of the ideal $\text{Fin}^3$.