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

摘要

We use a countable-support product of invariant Jensen's forcing notions to define a model of ZFC set theory in which the uniformization principle fails for some planar Pi(1)(2) 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 Pi(1)(2) sequence of Vitali classes P-n whose union boolean OR(n) P-n is not a countable set. Of course, the axiom of choice fails in this submodel.