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

Abstract

We present a model of set theory, in which, for a given $n\ge 2$, there exists a non-ROD-uniformizable planar lightface ${\Pi }_n^1$ set in $\mathbb{R}\times \mathbb{R}$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface ${\mathbf{\Sigma }}_{\mathbf{n}}^1$ sets with countable cross-sections are ${\mathbf{\Delta }}_{\mathbf{n}+1}^1$-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.

A revised version of the originally submitted preprint

References in corpus (1)

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

Cited by in corpus (4)

• Definable minimal collapse functions at arbitrary projective levels
• Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
• On intermediate extensions of generic extensions by a random real
• A product forcing model in which the Russell-nontypical sets satisfy ZFC strictly between HOD and the universe