Brouwer’s Equivalence Between Virtual and Inextensible Order

  • Enrico MartinoEmail author
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 42)


Brouwer’s theorem of 1927 on the equivalence between virtual and inextensible order is discussed. Several commentators considered the theorem at issue as problematic in various ways. Brouwer himself, at a certain time, believed to have found a very simple counterexample to his theorem. In some later publications, however, he stated the theorem in the original form again. It is argued that the source of all criticisms is Brouwer’s overly elliptical formulation of the definition of inextensible order , as well as a certain ambiguity in his terminology. Once these drawbacks are removed, his proof goes through.


  1. Brouwer, L. (1927). Virtuelle ordnung und unerweiterbare ordnung. Journal für die reine und angewandte Mathematik, 157, 255–257, also in Brouwer (1975, pp. 406–408).Google Scholar
  2. Brouwer, L. (1950). Remarques sur la notion d’ordre. Comptes Rendus de l’Académie des Sciences de Paris, 230, 263–265, also in Brouwer (1975, pp. 499–500).Google Scholar
  3. Brouwer, L. (1975). Collected works. In A. Heyting (Ed.), Philosophy and foundations of mathematics. Amsterdam: Elsevier.Google Scholar
  4. Brouwer, L. (1981). In van Dalen, D. (Ed.), Cambridge lectures in intuitionism. Cambridge: Cambridge University Press.Google Scholar
  5. Heyting, A. (1975). Notes in Brouwer. Amsterdam: Elsevier.Google Scholar
  6. Posy, C. J. (1980). On Brouwer’s definition of unextendable order. History and Philosophy of Logic, 1, 139–149.CrossRefGoogle Scholar
  7. van Dalen, D. (1981). Notes in Brouwer. Cambridge: Cambridge University Press.Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.FISPPA DepartmentUniversity of PaduaPaduaItaly

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