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arXiv · 1712.00769

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

Vladimir Kanovei·Vassily Lyubetsky
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Abstract

We present a model of set theory, in which, for a given $n\ge2$, there exists a non-ROD-uniformizable planar lightface $\varPi^1_n$ set in $\mathbb R\times\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\bfΣ^1_n$ sets with countable cross-sections are $\bfΔ^1_{n+1}$-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.

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Vladimir Kanovei, Vassily Lyubetsky. 2018-01-31. Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy. https://doi.org/10.1070/im8521

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