Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes

Abstract

We use a countable-support product of invariant Jensen's forcing notions to define a model of set theory in which the uniformization principle fails for some planar set all of whose vertical cross- sections are countable sets and, more specifically, Vitali classes. We also define a submodel of that model, in which there exists a countable sequence of Vitali classes whose union is not a countable set. Of course, the axiom of choice fails in this submodel.