On Farah ideals and beyond

An ideal I on $ω$ is called Farah if there are hereditary closed subsets Cₙ of P(ω), for n∈ω, such that I=⊆ω:\ ∀n∈ω\ ∃k∈ω\ A k∈ Cₙ\. Farah asked in [Trans. Amer. Math. Soc. 356 (2004), no. 6] if every Fσδ ideal is a Farah ideal. We show that this is false by providing both tall and non-tall counterexamples. Moreover, we show that there is an antichain in the sense of Katětov order of length consisting of Fσδ non-Farah ideals. We also investigate a closely related problem concerning sets of ideal limit points. We show that if I is an Gδσδ ideal on $ω$ and x∈ X^ω is a sequence in an uncountable Polish space X, then \[ Λ_x(I)=\{p∈ X:\ ∃A\ x A converges to p\} \] is F_σ. This answers [Top. App. 312 (2022)] (partially) and [J. Symb. Logic, DOI: 10.1017/jsl.2026.10181]. Earlier it was known to be true only in the case of Farah ideals.