Bases and selectors for tall families

TL;DR

This paper explores the limitations of certain theorems and constructs specific examples of tall ideals with particular properties, revealing differences in uniformity and definability.

Contribution

It demonstrates the existence of a uniform version of Nash-Williams theorem and constructs tall ideals with unique definability and selector properties.

Findings

Nash-Williams theorem has a uniform version, Galvin theorem does not

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

Construction of a $oldsymbol\Pi^1_2$ tall ideal without a tall closed subset

Abstract

We show that the Nash-Williams theorem has a uniform version and that the Galvin theorem does not. We show that there is an $F_\sigma$ tall ideal on $\mathbb{N}$ without a Borel selector and also construct a $\mathbf\Pi^1_2$ tall ideal without a tall closed subset.