Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy

TL;DR

This paper constructs a set-theoretic model demonstrating that the uniformization principle for sets with countable cross-sections fails at a specific projective level, highlighting a precise boundary in the hierarchy.

Contribution

It introduces a model where non-ROD-uniformizable sets with countable cross-sections exist at a given projective level, clarifying the limits of uniformization principles.

Findings

Existence of a non-ROD-uniformizable $oldsymbol{oldsymbol{ extPi}^1_n}$ set with countable cross-sections.

All $oldsymbol{foldsymbol{ extSigma}^1_n}$ sets with countable cross-sections are $oldsymbol{oldsymbol{ extDelta}^1_{n+1}}$-uniformizable.

The uniformization principle fails exactly at the specified projective level in the constructed model.

Abstract

We present a model of set theory, in which, for a given $n\ge2$, there exists a non-ROD-uniformizable planar lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\bf\Sigma^1_n$ sets with countable cross-sections are $\bf\Delta^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.