Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
Volume 245 / 2019
Fundamenta Mathematicae 245 (2019), 175-215
MSC: Primary 03E35; Secondary 03E15.
DOI: 10.4064/fm517-7-2018
Published online: 28 January 2019
Abstract
We present a model of set theory in which, for a given $n\ge 2$, there exists a planar non-ROD-uniformizable lightface $\varPi ^{1}_{n}$ set, all of whose vertical cross-sections are countable sets and, more specifically, Vitali classes, while all planar boldface ${\mathbf \Sigma }^{1}_{n}$ sets with countable cross-sections are ${\mathbf \Delta }^{1}_{n+1}$-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.
