Constructivism in Mathematics, Vol 1
Published on 1. July 1988
355 pages
978-0-08-057088-4 (ISBN)
Description
These two volumes cover the principal approaches to constructivism in mathematics. They present a thorough, up-to-date introduction to the metamathematics of constructive mathematics, paying special attention to Intuitionism, Markov's constructivism and Martin-Lof's type theory with its operational semantics. A detailed exposition of the basic features of constructive mathematics, with illustrations from analysis, algebra and topology, is provided, with due attention to the metamathematical aspects. Volume 1 is a self-contained introduction to the practice and foundations of constructivism, and does not require specialized knowledge beyond basic mathematical logic. Volume 2 contains mainly advanced topics of a proof-theoretical and semantical nature.
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Additional editions
A.S. Troelstra | D. van Dalen
Constructivism in Mathematics, Vol 1: Volume 121
Book
07/1988
Elsevier
€195.99
Shipment within 15-20 days
Content
- Front Cover
- Constructivism in Mathematics: An Introduction
- Copyright Page
- CONTENTS OF VOLUME I
- Preface
- Preliminaries
- Chapter 1 Introduction
- 1. Constructivism
- 2. Constructivity
- 3. Weak counterexamples
- 4. A brief history of constructivism
- 5. Notes
- Exercises
- Chapter 2 Logic
- 1. Natural deduction
- 2. Logic with existence predicate
- 3. Relationships between classical and intuitionistic logic
- 4. Hilbert-type systems
- 5. Kripke semantics
- 6. Completeness for Kripke semantics
- 7. Definitional extensions
- 8. Notes
- Exercises
- Chapter 3 Arithmetic
- 1. Informal arithmetic and primitive recursive functions
- 2. Primitive recursive arithmetic PRA
- 3. Intuitionistic first-order arithmetic HA
- 4. Algorithms
- 5. Some metamathematics of HA
- 6. Elementary analysis and elementary inductive definitions
- 7. Formalization of elementary recursion theory
- 8. Intuitionistic second-order logic and arithmetic
- 9. Higher-order logic and arithmetic
- 10. Notes
- Exercises
- Chapter 4 Non-classical axioms
- 1. Preliminaries
- 2. Choice axioms
- 3. Church's thesis
- 4. Realizability
- 5. Markov's principle
- 6. Choice sequences and continuity axioms
- 7. The fan theorem
- 8. Bar induction and generalized inductive definitions
- 9. Uniformity principles and Kripke's schema
- 10. Notes
- Exercises
- Chapter 5 Real numbers
- 1. Introduction
- 2. Cauchy reals and their ordering
- 3. Arithmetic on R
- 4. Completeness properties and relativization
- 5. Dedekind reals
- 6. Arithmetic of Dedekind reals and extended reals
- 7. Two metamathematical digressions
- 8. Notes
- Exercises
- Chapter 6 Some elementary analysis
- 1. Intermediate-value and supremum theorems
- 2. Differentiation and integration
- 3. Consequences of WC-N and FAN
- 4. Analysis in CRM: consequences of CT0, ECT0, MP
- 5. Notes
- Exercises
- Bibliography
- Index
- List of symbols
