N. N. Luzin's problems on the existence of CA-sets without perfect subsets
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 \(V=L.\)
scientific article; zbMATH DE number 4041907
scientific article