My question is about terminology:
Do you know why stationary sets were named such?
Going over the following MO question about the intuition behind stationary sets, the only compelling argument I can think of is Fodor's lemma.
Is this the reason?
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5$\begingroup$ Ioannis, this was addressed in MSE. They were also referred early on as Mahlo sets. $\endgroup$– Andrés E. CaicedoCommented Jul 26, 2013 at 0:16
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$\begingroup$ @Andres: Thank you for the link. It indeed answers the question. I never seem to look MSE for some reason. $\endgroup$– Ioannis SouldatosCommented Jul 26, 2013 at 13:39
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1 Answer
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In Infinite Combinatorics, in: Handbook of the History of Logic, 6. Sets and Extensions in the Twentieth Century, p 226, footnote 214, Jean Larson states that the term was first used in G. Bloch: Sur les ensembles stationnaires de nombres ordinaux et les suites distinguees de fonctions regressives, Comptes Rendus Acad. Sci Paris, 236(1953), 265-268. The reason for the name was probably Neumer's theorem, a weaker and earlier form of Fodor's theorem.
answered Jul 26, 2013 at 6:23
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$\begingroup$ Indeed. Neumer's theorem states that if $S$ is a stationary subset of a cardinal $\rho$ of uncountable cofinality, and $f:S\to\rho$ is regressive, then $f$ is bounded on a cofinal set, that is, there is an $\alpha<\rho$, and a $D\subset S$, $D$ cofinal, such that $f(\beta)<\alpha$ for all $\beta\in D$. The reference is W. Neumer. Verallgemeinerung eines Satzes von Alexandroff und Urysohn, Math. Z., 54, (1951), 254—261. MR0043860 (13,331a). $\endgroup$ Commented Jul 26, 2013 at 6:36
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$\begingroup$ Actually, Bloch's precise statement is a bit more cumbersome than I mentioned. He uses a notion similar to club, that he calls a band: Instead of requiring that $C\subset\rho$ is cofinal, he requires that it has order type $\rho$. Such $C$ he calls full parts, and bands are sets that are closed full parts. The theorem mentioned in the title of Neumer's paper is a particular case of a result of Dushnik from the early 1930s. (The version I stated above is a modern reformulation. Juhász proved in the 70s that it is essentially a topological result.) $\endgroup$ Commented Jul 26, 2013 at 6:40
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$\begingroup$ Hmm... Bloch closes his note mentioning Neumer's paper, but what it says is: "Cet article ne m'était pas connu quand j'ai rédigé la présente Note." Other than this, Bloch's article includes no references. $\endgroup$ Commented Jul 26, 2013 at 7:04
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4$\begingroup$ @Andres: So these bands, do they play in any clubs? Or do they move from town to town too fast for such gigs? (i.e. are they non-stationary?) :-) $\endgroup$– Asaf Karagila ♦Commented Jul 26, 2013 at 9:37