N. N. Luzin's problems on the existence of CA-sets without perfect subsets
type: Item
instance of: scholarly article
title: N. N. Luzin's problems on the existence of CA-sets without perfect subsets
zbMATH Open document ID: 0639.03055
DOI: 10.1007/BF01159870
published in: Mathematical Notes
publication date: 1987-01-01T00:00:00Z
review text: The paper concerns three classical problems of descriptive set theory, stated by N. Luzin: problems of the existence of: (i) a CA-set of cardinality strictly between \(\aleph_ 0\) and \({\mathfrak c}\), (ii) a function whose graph is a CA-set without perfect subsets, (iii) an uncountable CA-set without any perfect subset. As is known from works by \textit{R. Solovay} [Ann. Math., II. Ser. 92, 1-56 (1970; Zbl 0207.009); Found. Math., Symp. Pap. Commem. 60th Birthday K. Gödel, Columbus 1966, 58-73 (1969; Zbl 0188.325)] and \textit{P. S. Novikov} [Tr. Mat. Inst. Steklova 38, 279-316 (1951; Zbl 0053.364)], these problems are insolvable within ZFC set theory. Owing to the known implications between the conjectures stated above, and the Continuum Hypothesis, there are only five logical combinations of them, namely: (1) \(\neg\) (iii) \(\wedge CH\), (2) \(\neg\) (iii) \(\wedge \neg CH\), (3) (ii), (4) (iii) \(\wedge \neg CH\), (5) (iii) \(\wedge \neg\) (ii) \(\wedge CH.\) The consistency of statements (1)-(4) was established in the quoted papers of Solovay and Novikov. The author of the present paper completes this picture giving the proof of the consistency of (5). The model of (5) built up here is \(\omega_ 1\)-Cohen generic extension of a transitive countable model of \(V=L.\)
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 4041907
zbMATH Keywords: CA-set
zbMATH Keywords: Continuum Hypothesis
zbMATH Keywords: consistency
author: Vladimir Kanovei
MaRDI profile type: Publication
cites work: Q5819572
cites work: A model of set-theory in which every set of reals is Lebesgue measurable
cites work: On the Cardinality of $$ \sum_2^1 $$ Sets of Reals
cites work: Perfect subsets of definable sets of real numbers
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description: scientific article; zbMATH DE number 4041907