Developments from Ernst Specker's work in set theory
type: Item
instance of: scholarly article
title: Developments from Ernst Specker's work in set theory
zbMATH Open document ID: 1254.03084
DOI: 10.4171/EM/200
author: Andreas Blass
published in: Elemente der Mathematik
publication date: 2012-09-21T00:00:00Z
reviewed by: T. Thrivikraman
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 03-03
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 6084847
zbMATH Keywords: Specker lines
zbMATH Keywords: history of set theory
MaRDI profile type: Publication
cites work: Cardinal characteristics and the product of countably many infinite cyclic groups
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cites work: Consistency results about filters and the number of inequivalent growth types
cites work: THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS
cites work: Q3995749
cites work: Some counterexamples in the partition calculus
cites work: Martin's axiom implies the existence of certain slender groups
cites work: The axiom of choice
cites work: A five element basis for the uncountable linear orders
cites work: Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip
cites work: On the axiom of determinateness
cites work: Verallgemeinerung eines Satzes von Herrn F. Specker
cites work: Decomposing uncountable squares to countably many chains
cites work: Q4753168
cites work: A model of set-theory in which every set of reals is Lebesgue measurable
cites work: Sur un problème de Sikorski
cites work: Q5802193
cites work: Zur Axiomatik der Mengenlehre (Fundierungs‐ und Auswahlaxiom)
cites work: Partitioning pairs of countable ordinals
cites work: The theory of semisets
review text: In this note, the author gives an account of Ernst Specker's work in set theory -- restricting himself to three of Specker's papers. The first one is the paper [\textit{E. Specker}, ``Sur un problème de Sikorski'', Colloq. Math. 2, 9--12 (1949; Zbl 0040.16703)]. An unpublished work of Specker closely related to this paper showed the existence of Specker lines -- that is, linearly ordered sets of cardinality \(\aleph_1\) such that no uncountable subset is well-ordered, anti-well-ordered or embeddable into the real line \(\mathbb R\) with the usual order. The existence of Specker lines was the key to the proof by \textit{F. Galvin} and \textit{S. Shelah} in [J. Comb. Theory, Ser. A 15, 167--174 (1973; Zbl 0267.04006)] of the negative partition relation, namely, one can partition the set of 2-element subsets of a set \(A\) of size \(\aleph_1\) into four pieces in such a way that every uncountable subset of \(A\) contains pairs from all four pieces. \textit{S. Shelah} solved in [J. Comb. Theory, Ser. A 21, 110--114 (1976; Zbl 0366.04009)] a long-standing problem by proving the existence of a Countryman line, i.e., a linear order \(L\) of cardinality \(\aleph_1\) such that \(L \times L\) with componentwise partial order is the union of countably many chains. Clearly every Countryman line is a Specker line. The work of \textit{J. T. Moore} in [Ann. Math. (2) 163, No. 2, 669--688 (2006; Zbl 1143.03026)] is also related.NEWLINENEWLINEThe second paper is [\textit{E. Specker}, ``Additive Gruppen von Folgen ganzer Zahlen'', Port. Math. 9, 131--140 (1950; Zbl 0041.36314)], which contains some surprising properties of the additive group \(\mathbb Z^N\) of all sequences of integers and its `monotone' subgroups.NEWLINENEWLINEIn Part II of the third paper [``Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom)'', Z. Math. Logik Grundlagen Math. 3, 173--210 (1957; Zbl 0079.07605)], \textit{E. Specker} gives a reformulation and abstraction of the Fraenkel-Mostowski method of…
description: scientific article; zbMATH DE number 6084847