Logical Number Theory I An Introduction

Number theory as studied by the logician is the subject matter of the book. This first volume can stand on its own as a somewhat unorthodox introduction to mathematical logic for undergraduates, dealing with the usual introductory material: recursion theo

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Craig Smorynski

Logical Number Theory I An Introduction

With 13 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Craig Smorytlski 429 S.Warwiek Westmont, IL 60559, USA

Cover picture: It does' not seem supererogatory to inform the reader that the cover illustration is derived from the frontispiece to the first edition of Christopher Marlowe's biography of Dr. Johannes Faustus, whose insightful remarks on logie are quoted in this book.

Mathematics Subject Classification (1980): 03-01, 03F30, 10B99, lON05

ISBN-I3: 978-3-540-52236-2 DOl: 10.1007/978-3-642-75462-3

e-ISBN-13: 978-3-642-75462-3

Library of Congress Cataloging-in-Publication Data. Smorynski, C. Logical number theory: an introduction/Craig Smoryliski. p. cm. - (Universitext) Includes bibliographical references and indexes. ISBN 3-540-52236-0 (SpringerVerlag Berlin: v. 1). - ISBN 0-387-52236-0 (Springer-Verlag New York: v. 1) 1. Number theory. 2. Logic, Symbolic and mathematical. I. Title. QA241.S614 1991 512'.7-dc20 90-25702 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 4113140-543210 - Printed on acid-free paper

Preface

What would life be like without arithmetic, but a

scene of honors? -

Rev. Sydney Smith

It is with a mixture of trepidation and chutzpah that I offer a work bearing the audacious title of Logical Number Theory. Disadvantageous comparisons with the spectacularly successful analytic number theory are inevitable. It is a fact, however, that logicians have made their own studies of arithmetic, that, although not as deep or as fully developed as analytic number theory, there is a body of logical knowledge of the subject. The present work concerns itself with this fledgling logical study of number theory. The present work, of which this book is the frrst of two volumes, is an introductory text. This means several things. First, it means that certain advanced and non-central topics have not been covered. Adequate expositions of some of these subjects already exist, and others are in preparation. Chapter VB of the second volume will offer references and recommendations to the reader who wishes to go further. For the present, let it suffice for me to say that these two volumes are intended to present the core of logical number theory and they do not go much beyond the basics. (I have, of course, indulged myself occasionally and included some items not clearly (or: clearly not) of central importance. I have done this whenever I felt the in