Erdős, Pál; Hajnal, András Some remarks concerning our paper ’On the structure of set-mappings’. Non-existence of a two-valued \(\sigma\)-measure for the first uncountable inaccessible cardinal. (English) Zbl 0134.01602 Acta Math. Acad. Sci. Hung. 13, 223-226 (1962). A cardinal \(m\) has property \(P_3\) if every two-valued measure \(\mu\) defined on the power set of a set \(S\) of power \(m\) is identically zero, provided that \(\mu (x)\) is \(m\)-additive and \(\mu(\{x\}) = 0\) for all \(x \in S\). [Cf. Erdős and Tarski, Essays Foundations Math., dedicat, to A. A. Fraenkel on his 70th Anniversary, 50-82 (1962; Zbl 0212.32502).] In the authors’ previous paper [Acta Math. Acad. Sci. Hung. 9, 111–131 (1958; Zbl 0102.28401)], they proved: (i) If \(m > \aleph_0\) is strongly inaccessible and does not have property \(P_3\), then \(m \to (m)^{<\aleph_0}\). (ii) \(m \not\to (\aleph_0)^{\aleph_0}\) for every \(m < t_1\), where \(t_1\) is the first uncountable strongly inaccessible ordinal. The partition notation \(m \to (n)^{\aleph_0}\) comes from P. Erdős and R. Rado [Bull. Am. Math. Soc. 62, 427–489 (1956; Zbl 0071.05105)]. They now derive from (ii) the additional result (iii): \(t_1 \not\to (\aleph_1)^{\aleph_0}\). From (i) and (iii) it follows that \(t_1\) has property \(P_3\), which had already been proved by Tarski and by Keisler. The authors state the following generalization of (iii): (iv) If \(n\) is either \(\aleph_0\) or not strongly inaccessible and \(t_\xi\) is the least strongly inaccessible ordinal \(>n\), then \(t_\xi \not\to (n^+)^{<\aleph_0}\). If \(t_0,t_1,...,t_\xi,...\) is an enumeration of all strongly inaccessible cardinals, then (i) and (iv) imply that, if \(\xi < t_\xi\) has \(P_3\). Among the unsolved problems mentioned, two of the simplest are: \(t_{\xi_0} \not\to (t_{\xi_0})^{\aleph_0}\) (where \(\xi_0\) is the least ordinal for which \(\xi_0 = t_\xi\)), and \(t_1 \to (\aleph_0)^{\aleph_0}\). Reviewer: E.Mendelsohn Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 7 Documents MSC: 05D10 Ramsey theory 03E55 Large cardinals 03E05 Other combinatorial set theory 05E05 Symmetric functions and generalizations 03E10 Ordinal and cardinal numbers Keywords:set theory Citations:Zbl 0212.32502; Zbl 0102.28401; Zbl 0071.05105 PDFBibTeX XMLCite \textit{P. Erdős} and \textit{A. Hajnal}, Acta Math. Acad. Sci. Hung. 13, 223--226 (1962; Zbl 0134.01602) Full Text: DOI References: [1] P. Erdos andA. Hajnal, On the structure of set-mappings,Acta Math. Acad. Sci. Hung.,9 (1958), pp. 111–131. · Zbl 0102.28401 [2] A. Tarski, Some problems and results relevant to the foundations of set theory,Proceedings of the International Congress for Logic, Methodology and Philosophy of Science (Stanford, 1960). [3] H. J. Keisler, Some applications of the theory of models to set theory,Proceedings of the International Congress for Logic, Methodology and Philosophy of Science (Stanford, 1960). · Zbl 0107.00803 [4] P. Erdos andA. Tarski, On some problems involving inaccessible cardinals,Esseys on the foundations of mathematics. Magnes Press. The Hebrew University of Jerusalem (1961), pp. 50–82. [5] P. Erdos andR. Rado, A partition calculus in set theory,Bull. Amer. Math. Soc.,62 (1956) pp. 427–489. · Zbl 0071.05105 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.