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

Vladimir Kanovei, Vassily Lyubetsky

Published 2017-12-03Version 1

A generic extension of $L$, the constructible universe, is defined, in which it is true for a given $n\ge2$ that there exists a non-ROD-uniformizable lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$ with all vertical cross-sections being Vitali classes (hence, countable sets of reals), and in the same time every boldface $\bf\Sigma^1_n$ set with countable cross-sections is uniformizable by a boldface $\bf\Delta^1_{n+1}$ set. Thus it is true in this model that the ROD-uniformization for sets with countable cross-sections first fails exactly at a given projective level.