Elements of intuitionism. (English) Zbl 0358.02032

Dummett, Michael

Oxford Logic Guides. Oxford: Clarendon Press; Oxford University Press. XII, 467 p. £9.00 (1977).

The purpose of this valuable book is to give, in a systematic way, an outline of the fundamentals of intuitionistic logic and mathematics. Chapter 1 provides a brief introduction to the intuitionistic program. In Chapter 2, the basic concepts of the intuitionistic theory of real numbers are explained. Chapter 3 is devoted to the elements of the theory of choice sequences and spreads. The basic definitions and results concerning the syntax of intuitionistic logic are presented in Chapter 4. Next, Chapter 5 introduces the basic notions of the semantics of intuitionistic logic, with particular emphasis on Beth trees. The author examines the completeness problem of first-order intuitionistic predicate logic, taking into account the most recent results of the Nijmegen school. Chapter 6 investigates some further topics. Special reference is made to Kleene’s notion of realizability. In the concluding Chapter 7, the author discusses in detail the philosophical foundations of the intuitionistic attitude towards mathematical knowledge. This excellent book can be recommended to the student of mathematics or philosophy wishing to get a comprehensive and reliable introduction into modern intuitionism. It is intended for the study at an intermediate level and nothing more than some acquaintance with classical mathematical logic is presupposed.

Reviewer: D. Schwartz

MSC:

03F55	Intuitionistic mathematics
03-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
03-02	Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations