Developments from Ernst Specker's work in set theory
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 An unpublished work of Specker closely related to this paper showed the existence of Specker lines -- that is, linearly ordered sets of cardinality \(\aleph_1\) such that no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 permutation models for the negation of the axiom of choice. And in Part III, Specker analyzes various properties of ordinals that become possible when the axiom of choice is not assumed.
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