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Foundations of Constructive Mathematics

Metamathematical Studies

  • Book
  • © 1985

Overview

Authors:
  1. Michael J. Beeson

    1. Department of Mathematics and Computer Science, San Jose State University, San Jose, USA

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Table of contents (17 chapters)

  1. Front Matter

    Pages I-XXIII
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  2. Practice and Philosophy of Constructive Mathematics

    1. Front Matter

      Pages 1-1
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    2. Examples of Constructive Mathematics

      • Michael J. Beeson
      Pages 3-32

    3. Informal Foundations of Constructive Mathematics

      • Michael J. Beeson
      Pages 33-46

    4. Some Different Philosophies of Constructive Mathematics

      • Michael J. Beeson
      Pages 47-57

    5. Recursive Mathematics: Living with Church’s Thesis

      • Michael J. Beeson
      Pages 58-81

    6. The Role of Formal Systems in Foundational Studies

      • Michael J. Beeson
      Pages 82-92

  3. Formal Systems of the Seventies

    1. Front Matter

      Pages 93-93
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    2. Theories of Rules

      • Michael J. Beeson
      Pages 97-147

    3. Realizability

      • Michael J. Beeson
      Pages 148-161

    4. Constructive Set Theories

      • Michael J. Beeson
      Pages 162-201

    5. The Existence Property in Constructive Set Theory

      • Michael J. Beeson
      Pages 202-215

    6. Theories of Rules, Sets, and Classes

      • Michael J. Beeson
      Pages 216-245

    7. Constructive Type Theories

      • Michael J. Beeson
      Pages 246-283

  4. Metamathematical Studies

    1. Front Matter

      Pages 285-285
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    2. Constructive Models of Set Theory

      • Michael J. Beeson
      Pages 287-299

    3. Proof-Theoretic Strength

      • Michael J. Beeson
      Pages 300-323

    4. Some Formalized Metamathematics and Church’s Rule

      • Michael J. Beeson
      Pages 324-345

    5. Forcing

      • Michael J. Beeson
      Pages 346-367

    6. Continuity

      • Michael J. Beeson
      Pages 368-398
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Keywords

About this book

This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec­ tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con­ structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.

Authors and Affiliations

  • Department of Mathematics and Computer Science, San Jose State University, San Jose, USA

    Michael J. Beeson

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