Diffusion, Crisis, and Bifurcation: 1890 to 1914

  • José Ferreirós
Part of the Science Networks. Historical Studies book series (SNHS, volume 23)


The years up to 1914 were a crucial period of diffusion and recognition for set theory. During the 1890s the new vision of mathematics and the Cantorian ideas spread out, while the 1900s saw fundamental new contributions in the hands of a new generation of cultivators — Zermelo and Hausdorff above all. But this was also a period of heated debates surrounding the notion of arbitrary set and its expressions, the Axiom of Choice and the Well-Ordering Theorem. It was also the time in which the paradoxes emerged, heralded by Russell. Thus, the diffusion of set theory was accompanied by much ambivalence and confusion. The acceptability of abstract mathematics was in question, as were the relations between logic and set theory.


Axiom System Cardinal Number Propositional Function Inaccessible Cardinal Predicative Function 
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  1. 1.
    Julius König [1905] as translated in [van Heijenoort 1967, 145].Google Scholar
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    Hilbert tried to solve it twice unsuccessfully, in [Hilbert 1926] and two years later.Google Scholar
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    That can also be presented as a version of Cantor’s definition, according to which a cardinal number is a general concept under which equipollent classes fall, but again not faithfully. We can here observe how by 1900 set theory was not yet completely extensionalized.Google Scholar
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    The axiom is formulated for a family of disjoint sets in order to make it simple and more intuitive.Google Scholar
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    [Zermelo 1908], as translated in [van Heijenoort 1967, 201], with a small change to accommodate the word `Klassenaussage.’Google Scholar
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    The strange features of the famous Tractatus by his student Wittgenstein [1921] are thus more a symptom than a deviation.Google Scholar
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    Likewise, systems of propositional and predicate logic started to be presented in the now customary way during the 1920s. Elements of the distinction between syntax and semantics can be found in Peano and Schröder, and also in Frege, insofar as he differentiates clearly between a name and its referent, between use and mention of an expressionGoogle Scholar
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    [Hausdorff 1914, 1–2]. His confidence in the Zermelo system is even clearer in the second, very abridged edition [1927, 34].Google Scholar
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    The bibliography only cites [Dedekind 1872]; see also chapters 2 and 9, where he fails to mention Dedekind in connection with functions and mappings.Google Scholar
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    The emergence of topology was a very complex, many-sided process. Here we pay attention to developments in set-theoretic topology leading up to the fundamental notion of topological space; for further details see [Manheim 1964, chap. 6; Johnson 1979; 1981]. Aspects of the rise of combinatorial and algebraic topology are studied in [Bollinger 1972; vanden Eynde 1992; Epple 1995].Google Scholar
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    Weyl had made a similar contribution previously [1913], also stimulated by Hilbert’s proposal. Hausdorff claims that his work was independent and that he presented it in 1912 at the University of Bonn [1914, 456–57].Google Scholar
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    The argument emphasized the similarity between the theory of order types and that of topological spaces: an order relation can be taken to be a two-valued function of two arguments, and topology can be developed on the basis of a distance function.Google Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • José Ferreirós
    • 1
  1. 1.Dpto. de Filosofía y Lógica, Avda. San Francisco Javier, s/nUniversidad de SevillaSevillaSpain

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