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Zermelo’s Axiom and Axiomatization in Transition (1908–1918)

  • Gregory H. Moore
Part of the Studies in the History of Mathematics and Physical Sciences book series (HISTORY, volume 8)

Abstract

As early as 1896, even before the discovery of set-theoretic paradoxes, a few mathematicians had suggested that set theory ought to be axiomatized. Yet interest in formulating such an axiomatization remained very faint even in 1903, when Russell restated Burali-Forti’s result of 1897 as a paradox, and published his own paradox as well. Hilbert, for example, viewed Russell’s paradox as revealing that contemporary logic failed to meet the demands of set theory.1 Russell asserted further that a solution to the paradoxes would result only from a reappraisal of the assumptions used in logic, rather than from technical mathematics [1906, 37]. Unperturbed by the paradoxes, Zermelo concentrated on axiomatizing set theory within mathematics rather than on revising the underlying logical assumptions.

Keywords

Choice Function Propositional Function Transfinite Induction Hamel Basis Vicious Circle Principle 
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

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    Letter of 7 November 1903 to Frege, in Frege 1980, 51–52.Google Scholar
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    Zermelo 1908, 108, 117. For a given set M and a given choice function f on the family of all non-empty subsets of M, T is a θ-chain if (1) T is a set of subsets of M; (2) M is in T; (3) if A is in T, then A - {f(A)} is in T; (4) if S is a subset of T, then ⋂ S is in T.Google Scholar
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    Although the term maximal principles has become standard in the recent mathematical literature, a more appropriate designation would be extremal principles. Indeed, for many maximal principles there is a corresponding minimal principle asserting the existence of a minimal element, i.e., an element such that no other element is strictly smaller in the given partial order. For a history of maximal principles whose interpretation differs somewhat from the present book, see Campbell 1978.Google Scholar
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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Gregory H. Moore
    • 1
  1. 1.Department of Mathematics and Institute for History and Philosophy of Science and TechnologyUniversity of TorontoTorontoCanada

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