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Consolidation of Axiomatic Set Theory

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

Abstract

After 1918, most important contributions to the foundations of abstract set theory relied on modem axiom systems. But until the 1920s few authors adopted Zermelo’s axiom system explicitly.2 As we saw in the preceding chapter, many favored the theory of types because it seemed to offer a safer framework, and at the same time it was sufficient for the limited amount of set theory that is necessary in so-called classical mathematics. As late as 1939 Alonzo Church was writing that the simplified theory of types and Zermelo’s set theory were essentially similar, and the “safest cities of refuge” for classicist mathematicians at the time.3 But the Zermelo system had to compete with another alternative, the system of von Neumann, presented in 1925 and developed later by Bernays and Gödel (see §3).

Keywords

Type Theory Axiom System Universal Class Iterative Conception Generalize Continuum Hypothesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
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  2. 2.
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    The first to argue in earnest for Replacement, and to work out its consequences, was von Neumann in 1923. The history of Replacement has been carefully studied by Hallett [1984].Google Scholar
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    This point was emphasized by Hallett [1984].Google Scholar
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    Zermelo did not spell out the details of his logical standpoint, particularly the effect of second-order logic for the Skolem paradox. But see [Zermelo 1931] and [Grattan-Guinness 1979; Moore 1980; Dawson 1985a]. A clear discussion of the logical aspects can be found in [Shapiro 1991] or [Lavine 1994].Google Scholar
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    The `layers’ Q a of Zermelo are not cumulative, in contrast to his `sections’ P a [1930, 36]. Therefore it is the second which correspond to the ranks of the usual cumulative hierarchy; instead of P a we shall write V a for the ranks, as has become customary.Google Scholar
  18. 1.
    This term is taken from [Hellman 1989] and [Lavine 1994].Google Scholar
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    The same applies to different models with a single, common `basis’ of urelements (that differ in their `characteristic’). Likewise, of two different models with the same `characteristic,’ one is always isomorphic to a subdomain of the other [1930, 42].Google Scholar
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  21. 1.
    Indeed, a double sequence of models, since we also have to take into account the many possible `bases’ of urelements [Zermelo 1930, 42, 47].Google Scholar
  22. 2.
    Gödel had already presented this conception in conferences during the 1930s; see volume 3 of his Collected Works. Google Scholar
  23. 1.
    Strictly speaking, it is wrong to say that Gödel offered the iterative conception as justification of the ZFC axioms. This was done by later authors. See [Klaua 1964], [Kreisel 1965], [Shoenfield 1967], [Boolos 1971], [Wang 1974], [Parsons 1977].Google Scholar
  24. 2.
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    [Meschkowski 1967, 290]: “Eine Menge ist dann und nur dann `zu gross’, wenn sie der Menge aller Dinge aequivalent ist.” In this letter, von Neumann employed naive terminology, as he would keep doing later, but it is clear that he was aware of the dangers.Google Scholar
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    In his opinion the development of the system became much simpler this way [1928a, 346; 1929, 494].Google Scholar
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    This is the “class theorem” proved in the first installment of Bernays’ series of papers, nowadays called theorem of predicative existence of classes.Google Scholar
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    In the 60s, Kripke, Cohen and Solovay, working independently, established that NBG with Global Choice is also a conservative extension of ZFC. The result was published later by Feigner [1971].Google Scholar
  31. 1.
    We shall not enter into more details regarding NBG here. Readers interested in a detailed analysis may turn, e.g., to [Fraenkel, Bar-Hillel & Levy 1973]. I would also like to emphasize that the historical interplay between ZFC and NBG ought to be the subject of more detailed research.Google Scholar
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    He refrained from publishing immediately, since he hoped to get further results regarding the independence of AC and CH (see Moore in [Gödel 1990, 158]).Google Scholar
  33. 1.
    Shepherdson [1951/53]. It was a very special inner model: the minimal one that contains all the ordinals and is transitive, i.e., such that, whenever x is in the model, so are elements of X.Google Scholar
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  36. 1.
    As Solovay writes in [Gödel 1990, 8].Google Scholar
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  41. 1.
    As second-order logic can be seen to do, when the system is interpreted extensionally (§X.5).Google Scholar
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    I skip here over the issue of choosing between ZFC and NBG (see §3).Google Scholar
  43. 1.
    Another relevant but slightly earlier contribution is [Bemays 1935], which deals with Platonism in mathematics.Google Scholar
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  45. 1.
    See [Lakatos 1967], who calls this `quasi-empiricism,’ while I would prefer to speak of a hypothetical conception of (large parts of) mathematics. The most important popularizer of this view among philosophers has been Quine; see his [1953].Google Scholar
  46. 2.
    For Bourbaki’s hypothetical conception, see [1949, 3]. Perhaps I should mention that Bourbaki was no real person, but a fictitious character invented by a group of French mathematicians, who published their joint work under that name.Google Scholar
  47. 1.
    When the first volume of his treatise was published, Bourbaki [1954] had changed his axiom system slightly. Some indication of the difficulties can be found in [Corry 1996].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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