Constructivism in mathematics. An introduction. Volume I
type: Item
instance of: scholarly article
title: Constructivism in mathematics. An introduction. Volume I
zbMATH Open document ID: 0653.03040
author: A. S. Troelstra
author: Dirk van Dalen
published in: Studies in Logic and the Foundations of Mathematics
publication date: 1992-09-17T00:00:00Z
review text: In the preface the authors present this first volume as an all-round introduction to constructivism in the wide sense, covering in particular Brouwer's intuitionism, Bishop's constructive mathematics and Markov's constructive recursive mathematics. Its main concern is with these different forms of constructive mathematics and with the metamathematics of constructivism. After an introduction on constructivity and its history there follows a chapter on logic, treating natural deduction, classical and intuitionistic logic, Hilbert-type systems and Kripke semantics. The third chapter deals with primitive recursive and intuitionistic first order arithmetic, algorithms, inductive definitions, metamathematics of Heyting's arithmetic, higher order logic and arithmetic and the formalization of elementary recursion theory. Essential for a deeper understanding of the different forms of constructive mathematics are the principles discussed in chapter four, such as axioms of choice, continuity principles, Church's thesis, Markov's principle, fan theorem, bar induction, Kripke's schema. The remaining chapters 5 and 6 introduce constructive Cauchy and Dedekind reals, develop their arithmetic and part of elementary real analysis. All chapters contain a set of notes with supplementary information such as historical notes and suggestions for further reading as well as a substantial set of exercises. The book is very rich in content: it discusses about twenty formal systems and a multiple of special axioms. Generally the treatment is efficient and clear, although sometimes a little hasty. It introduces well into recent results and current research. It is a good guide to more specialized treatises on constructive (meta-) mathematics (among which Vol. II). However a book of about 300 pages, with an equal amount hidden in the exercises is not exactly an introduction for a reader with only some knowledge of classical first order predicate logic, as stated in the preface. Obviously it is a…
Mathematics Subject Classification ID: 03F50
Mathematics Subject Classification ID: 03F55
Mathematics Subject Classification ID: 03F60
Mathematics Subject Classification ID: 03F65
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 03-01
zbMATH DE Number: 41174
zbMATH Keywords: Brouwer's intuitionism
zbMATH Keywords: Bishop's constructive mathematics
zbMATH Keywords: Markov's constructive recursive mathematics
zbMATH Keywords: metamathematics of constructivism
reviewed by: B. van Rootselaar
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 41174