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

DOI10.4064/FM517-7-2018zbMATH Open1472.03059arXiv1712.00769OpenAlexW2963332506MaRDI QIDQ5376616

Vladimir Kanovei, Vassily Lyubetsky

Publication date: 13 May 2019

Published in: Fundamenta Mathematicae (Search for Journal in Brave)

Abstract: We present a model of set theory, in which, for a given $nge2$, there exists a non-ROD-uniformizable planar lightface $varP{i}_n^1$ set in $mathbbRimesmathbbR$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $$ sets with countable cross-sections are $$-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.

Full work available at URL: https://arxiv.org/abs/1712.00769

zbMATH Keywords

forcinguniformizationVitali class

Mathematics Subject Classification ID

Descriptive set theory (03E15) Consistency and independence results (03E35)

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Related Items (7)

Models of set theory in which the separation theorem fails ⋮ On the ‘definability of definable’ problem of Alfred Tarski, Part II ⋮ The full basis theorem does not imply analytic wellordering ⋮ A good lightface \(\varDelta_n^1\) well-ordering of the reals does not imply the existence of boldface \(\mathbf{\Delta}_{n - 1}^1\) well-orderings ⋮ Factoring Solovay-random extensions, with application to the reduction property ⋮ An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited ⋮ Ordinal definability and combinatorics of equivalence relations

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