Tall $F_\sigma$ subideals of tall analytic ideals

TL;DR

This paper proves that every analytic tall ideal on natural numbers contains an $F_\sigma$ tall ideal and provides an example of an $F_\sigma$ tall ideal lacking a Borel selector, advancing the understanding of ideal structures.

Contribution

It establishes the existence of $F_\sigma$ tall ideals within all analytic tall ideals and presents a counterexample regarding Borel selectors, answering a question by Hrušák.

Findings

Every analytic tall ideal contains an $F_\sigma$ tall ideal.

Existence of an $F_\sigma$ tall ideal without a Borel selector.

Addresses a question posed by Hrušák.

Abstract

Answering a question of Hru\v{s}\'ak, we show that every analytic tall ideal on $\omega$ contains an $F_\sigma$ tall ideal. We also give an example of an $F_\sigma$ tall ideal without a Borel selector.