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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 1, Pages 65–96 (Mi izv8521)  

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

V. G. Kanoveiab, V. A. Lyubetskyac

a Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
b Russian University of Transport
c Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We use a countable-support product of invariant Jensen's forcing notions to define a model of $\mathbf{ZFC}$ set theory in which the uniformization principle fails for some planar $\varPi_2^1$ 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 $\varPi_2^1$ sequence of Vitali classes $P_n$ whose union $\bigcup_nP_n$ is not a countable set. Of course, the axiom of choice fails in this submodel.

Keywords: uniformization, forcing, Vitali classes.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00705
Russian Science Foundation 14-50-00150
The first author acknowledges the support of RFBR grant no. 17-01-00705, and the second that of Russian Science Foundation grant no. 14-50-00150.


DOI: https://doi.org/10.4213/im8521

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English version:
Izvestiya: Mathematics, 2018, 82:1, 61–90

Bibliographic databases:

UDC: 510.225+510.223
MSC: 03E15, 03E35, 03E20
Received: 10.02.2016

Citation: V. G. Kanovei, V. A. Lyubetsky, “Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes”, Izv. RAN. Ser. Mat., 82:1 (2018), 65–96; Izv. Math., 82:1 (2018), 61–90

Citation in format AMSBIB
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