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Cylinder problem. (English) Zbl 0633.03044

If F is a family of subsets of a set E, closed under complements, [F] denotes the \(\sigma\)-algebra generated by F. If X is a set, \(n\leq m<\omega\) and \(i_ 0<...<i_{n-1}<m\), let \(C^ m_{\{i_ 0,...,i_{n-1}\}}(X)\) denote the family of all sets of the form \(\{<x_ 0,...,x_{m-1}>\in^ mX:\) \(<x_{i_ 0},...,x_{i_{n-1}}>\in S\}\) where \(S\subset^ nX\), and let \(C^ m_ n(X)=\cup \{C^ m_{\{i_ 0,...,i_{n-1}\}}(X):\) \(i_ 0<...<i_{n-1}<m\}\). For \(1\leq n\leq m<\omega\), let \(P^ m_ n(X)\) denote the sentence \({\mathcal P}(^ mX)=[C^ m_ n(X)]\), and let \(P_ n(X)\) stand for \(P_ n^{n+1}(X)\). The sentence \(P_ n(X)\) is called the \((n+1)\)-dimensional cylinder problem for X. The authors consider the question for which cardinals \(\kappa\) the problem \(P_ n(\kappa)\) has a positive solution. Th. 1: Let \(1\leq n<\omega\) and let \(\lambda\) be a cardinal such that cf \(\lambda\neq \omega_ 1\). If \(P_ n(\alpha)\) holds for every \(\alpha <\lambda\) then \(P_{n+1}(\lambda)\). This implies for any cardinal \(\kappa\) and any positive integer n: If \(P_ n(\kappa)\) then \(P_{n+1}(\kappa^+)\). In particular \(P_ n(\omega_ n)\) holds. Th. 2: If \(1\leq n<\omega\) and \(P_ n(\kappa)\) holds then \(\kappa \leq \beth_ n\). Hence, if GCH holds then for \(1\leq n<\omega\) we have \(P_ n(\kappa)\) iff \(\kappa \leq \omega_ n\). As a consequence of theorem 3 the authors obtain that \(P_ n(\beth_ n)\) is not a theorem of ZFC.
Reviewer: E.Harzheim

MSC:

03E15 Descriptive set theory
03E10 Ordinal and cardinal numbers
03E05 Other combinatorial set theory
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