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S. W. P. Steen, "Mathematical Logic with Special Reference to the Natural Numbers"

About ten years ago I conceived the idea of writing a book on the natural numbers because I thought that what had appeared up till then seemed to have reached a point where there was a certain amount of completeness - of course there never will be absolute completeness — and this is one of the attractions of the subject. Anyway it was not until I had retired that I had the time to get down to the task properly. The result is a book which begins with an account of formal languages including the two most basic, namely, the propositional calculus and the predicate calculus, and then goes on to arithmetic; beginning with a very simple arithmetic; finding this inadequate; extending it to overcome this inadequacy; finding the resulting system, though richer in modes of expression, still, but for a different reason, inadequate; extending this in turn to remedy this inadequacy; finding the resulting system has lost some of the ' nice' qualities of its predecessor, but is again, for a new reason, inadequate; extending this and so on. Before I come to develop arithmetic formally, it is convenient to have a primitive notation for the natural numbers (mainly to avoid lengthy circumlocutions) from which the concept of order and the operations of addition and multiplication can easily be obtained. I use sequences of tally marks, this is sufficient for our purposes. The real difficulty with arithmetic, as with other things, enters with the universal quantifier, when we want to make statements about all natural numbers. This use of tally marks is mentioned in the text but in the main is left to the reader to fill in. There are several topics absent from the book which might have been

included, these are partly off the main line of development, partly appli-applications of the general theory developed, partly sidelines, etc. Among these topics are: recursive analysis, constructive ordinals, recursive

equivalence types, recursive probability theory, the word problem, algorithms, finite automata, A-conversion, combinations, productions, intuitionism, various forms of propositional calculus, many-valued logics, and so on. Of these the constructive ordinals are mentioned several times because now and again we come across a process which can be continued into the constructible transfinite, but we do not go into it further.

The matter developed in this book was developed over the years in a course of lectures delivered at Cambridge, except that very little was said about the contents of Chs. 10, 11, 12, so these three chapters have not come under the fire of criticism of young scholars, and I feel that in consequence that they are not of the same quality as the earlier chapters, particularly the account of cut elimination in Ch. 10. The remaining chapters have been fairly well thrashed out in lecture and I am very appreciative of the comments of my classes and of the elegant onstra- demonstrations they gave me from time to time. I hope that I have acknowledged them all. With regard to the language in which the book is written, this is meant to consist of instructions and descriptions and occasionally of pointing out that such and such a procedure would lead to an impossible situation. Later in the book, when treating with ultra products I have transgressed and used Zorn's lemma, but a purist can tear that piece out of the book. Each chapter is followed by a short historical account of the matter treated in that chapter, it is this way that I make acknowledgement to those who first invented the matter, if I have made omissions then I apologize. After the historical account there follow a few examples. Many more examples can be found in books by Rogers A967), Shoenfeld A967) and Church A956). I must thank Professor R. Harrop and Dr N. Routledge for comments

on a former, now completely discarded draft which developed a much more complicated system. The present system owes its simplicity to the iterator symbol. I must also thank Dr G. T. Kneebone for reading the draft of Chs. 1-7 inclusive and providing valuable comments, and Drs T. J. Smiley and L. Drake for reading the draft of the remaining chapters and again providing valuable comments; also to the University Press for courtesy and consideration during the production of the book, and finally to my wife for help with the tedious business of making an index.

Introduction

Christ's College

Cambridge

June 1971

S.W.P.S.

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