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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 134.01602

**Autor: ** Erdös, Pál; Hajnal, András

**Title: ** 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 (In English)

**Source: ** Acta Math. Acad. Sci. Hung. 13, 223-226 (1962).

**Review: ** 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. *Erdös* and *Tarski*, Essays Foundations Math., dedicat, to A. A. Fraenkel on his 70th Anniversary, 50-82 (1962; Zbl 212.32502).] In the authors previous paper (Zbl 102.28401), they proved:

(i) If m > \aleph_{0} is strongly inaccessible and does not have property P_{3}, then m ––> (m)^{ < \aleph0}.

(ii) m \not ––> (\aleph_{0})^{\aleph0} for every m < t_{1}, where t_{1} is the first uncountable strongly inaccessible ordinal. The partition notation m ––> (n)^{\aleph0} comes from *P.Erdös* and *R.Rado* (Zbl 071.05105).

They now derive from (ii) the additional result (iii): t_{1} \not ––> (\aleph_{1})^{\aleph0}. 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 ––> (n^+)^{ < \aleph0}. 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_{\xi0} \not ––> (t_{\xi0})^{\aleph0} (where \xi_{0} is the least ordinal for which \xi_{0} = t_{\xi}), and t_{1} ––> (\aleph_{0})^{\aleph0}.

**Reviewer: ** E.Mendelsohn

**Classif.: ** * 05D10 Ramsey theory

03E55 Large cardinals

04A20 Combinatorial set theory

05E05 Symmetric functions

04A10 Ordinal and cardinal numbers; generalizations

**Index Words: ** set theory

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