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 P3 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 > \aleph0 is strongly inaccessible and does not have property P3, then m ––> (m) < \aleph0.
(ii) m \not ––> (\aleph0)\aleph0 for every m < t1, where t1 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): t1 \not ––> (\aleph1)\aleph0. From (i) and (iii) it follows that t1 has property P3, which had already been proved by Tarski and by Keisler. The authors state the following generalization of (iii):
(iv) If n is either \aleph0 or not strongly inaccessible and t\xi is the least strongly inaccessible ordinal > n, then t\xi \not ––> (n^+) < \aleph0. If t0,t1,...,t\xi,... is an enumeration of all strongly inaccessible cardinals, then (i) and (iv) imply that, if \xi < t\xi has P3. Among the unsolved problems mentioned, two of the simplest are: t\xi0 \not ––> (t\xi0)\aleph0 (where \xi0 is the least ordinal for which \xi0 = t\xi), and t1 ––> (\aleph0)\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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