RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

 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

Full text: PDF file (950 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

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

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
\Bibitem{KanLyu18} \by V.~G.~Kanovei, V.~A.~Lyubetsky \paper Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes \jour Izv. RAN. Ser. Mat. \yr 2018 \vol 82 \issue 1 \pages 65--96 \mathnet{http://mi.mathnet.ru/izv8521} \crossref{https://doi.org/10.4213/im8521} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2018IzMat..82...61K} \elib{http://elibrary.ru/item.asp?id=32428079} \transl \jour Izv. Math. \yr 2018 \vol 82 \issue 1 \pages 61--90 \crossref{https://doi.org/10.1070/IM8521} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000427245900004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85043703476}