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Self-Reference and Modal Logic

  • C. Smoryński

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Introduction

    1. C. Smoryński
      Pages 1-62
  3. The Logic of Provability

    1. Front Matter
      Pages N1-N1
    2. C. Smoryński
      Pages 63-86
    3. C. Smoryński
      Pages 87-132
    4. C. Smoryński
      Pages 133-165
  4. Multi-Modal Logic and Self-Reference

    1. Front Matter
      Pages N3-N3
    2. C. Smoryński
      Pages 217-254
  5. Non-Extensional Self-Reference

    1. Front Matter
      Pages N5-N5
    2. C. Smoryński
      Pages 255-297
    3. C. Smoryński
      Pages 298-329
  6. Back Matter
    Pages 330-333

About this book

Introduction

It is Sunday, the 7th of September 1930. The place is Konigsberg and the occasion is a small conference on the foundations of mathematics. Arend Heyting, the foremost disciple of L. E. J. Brouwer, has spoken on intuitionism; Rudolf Carnap of the Vienna Circle has expounded on logicism; Johann (formerly Janos and in a few years to be Johnny) von Neumann has explained Hilbert's proof theory-- the so-called formalism; and Hans Hahn has just propounded his own empiricist views of mathematics. The floor is open for general discussion, in the midst of which Heyting announces his satisfaction with the meeting. For him, the relationship between formalism and intuitionism has been clarified: There need be no war between the intuitionist and the formalist. Once the formalist has successfully completed Hilbert's programme and shown "finitely" that the "idealised" mathematics objected to by Brouwer proves no new "meaningful" statements, even the intuitionist will fondly embrace the infinite. To this euphoric revelation, a shy young man cautions~ "According to the formalist conception one adjoins to the meaningful statements of mathematics transfinite (pseudo-')statements which in themselves have no meaning but only serve to make the system a well-rounded one just as in geometry one achieves a well­ rounded system by the introduction of points at infinity.

Keywords

Arithmetic Calculation Logic addition algebra model theory proof proof theory theorem

Authors and affiliations

  • C. Smoryński
    • 1
  1. 1.Department of Mathematics and Computer ScienceSan Jose State UniversitySan JoseUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-8601-8
  • Copyright Information Springer-Verlag New York 1985
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-96209-2
  • Online ISBN 978-1-4613-8601-8
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site