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one saves precisely for such an occasion as a sudden request to contribute to one or another memorial volume. The four above classifications have non-empty intersections.

We are in favour of the idea of a memorial volume, in itself (for sentimental reasons, perhaps), but let the result also be really worthwhile in itself. In our opinion, this one fails to qualify on the last count.

Brno, Czechoslovakia JAN PASEKA Amsterdam, The Netherlands MICHIEL H A Z E W l N K E L

P. T. Johnstone: Notes on Logic and Set Theory (Cambridge Mathematical Text-

books), Cambridge University Press, 1987, paperback, l l0pp., £6.95/$12.95.

In nine short chapters, this book presents the basic facts and ideas from first-order logic and formal set theory: completeness proofs for propositional and predicate logic,

elements of recursive function theory, the ZF-axiomatization of set theory, ordinals, cardinals, and an informal introduction to independence proofs. It is meant to be used as lecture notes for a first (undergraduate) course in these subjects. Although undoubtedly clearly written, the presentation strikes me as being too succinct to provide a solid basis for such a course. Most students who are new to the subject will

run into many smaller and/or bigger difficulties that might be resolved in a classroom situation, but not (explicitly) by these notes.

Anyone trying to master the subject on his/her own, therefore, will sooner or later find it necessary to consult more comprehensive sources.

On the other hand, anyone teaching the subject will want to incorporate her/his own preferences (e.g. introduce first-order logic in a 'natural deduction style') which quickly renders substantial parts of such a small-scale work as the one under consideration, rather useless.

Therefore, as a textbook in logic and set theory, Dr Johnstone's book is not likely to become a companion to a great many courses. What remains is handy little reference book for anyone who has already done one or two courses in logic and set theory, but feels the need for a quick and pleasant refresher course.

Department of Mathematics and Computer Science, University of Amsterdam.

HAROLD SCHELLINX

J. R. Hindley and J. P. Seldin: Introduction to Combinators and A-Calculus (London Mathematical Society Student Texts 1), Cambridge University Press, 1986, paper- back, 360pp., £8.95/$16.95.

Combinatory logic (CL) and the related theory of the type-free A-calculus were introduced in the twenties and early thirties. An important original intention was to

Acta Applicandae Mathematicae 19 (1990)

BOOK REVIEWS 99

develop a theory that could provide a foundation for logic and mathematics. Due to the appearance of paradoxes, this ambitious plan could not be fulfilled. What is left is a theory in which the concepts of function application and (the inverse operation) of function abstraction are formalized. As opposed to the usual set-theoretical interpre- tation of a function as a subset of the Cartesian product of its domain and range, the 2- calculus stresses the operational aspects of functions, the idea of functions as rules for computation, as algorithms. In this capacity, 2-calculus possesses many of the characteristics of a programming language, which accounts for the attention paid to the theory in the realm of computer science, as well as for the many applications in both the theory and practice or programming.

The authors, Roger Hindley and Jonathan Seldin, both experts on the subject, give, in this book, an extensive introduction to techniques and basic results. The first half (Chapters 1-12) is devoted to the type-free theory. The fundamental notions for the 2- calculus and the corresponding ones for CL are introduced, and it is shown that all partial recursive functions are both 2 and combinatorially definable. By switching back and forth between 2-calculus and CL, from the very first pages the close correspondence between the two theories is stressed. This correspondence is made precise in Chapter 9, after which a neat introduction to the model theory of the ~.- calculus follows. A basic and detailed description of the construction of Scott's extensional Doo-model finishes the type-free part of the book.

The second half introduces theories that evolve through the assignment of types to 2- and CL terms (in 'Curry-style' as well as in 'Church-style') and gives a short overview of logics based on combinators. G6dels consistency proof for arithmetic through the interpretation of formulas as 3V-statements about primitive recursive functionals of finite type (the so-caUed Dialectica interpretation), is the subject of the final chapter.

There are many interesting topics in 2-calculus that cannot be found in this book, but then it does not claim to be a comprehensive introduction. The choices made by the authors are fair, the exposition is clear, and the presentation systematic. One can find ample historical notes and many useful references and suggestions for further study. Moreover, the authors only assume a very minimum of previous knowledge on the side of the reader. Some familiarity with predicate logic and the elements of recursive function theory should be sufficient to make a reading of this book a successful enterprise, both for students of mathematics and of computer science.

As a first introduction to the subject: highly recommended.

Department of Mathematics and Computer Science, University of Amsterdam.

HAROLD SCHELLINX