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Problems in , Mathematical and the Theory of THE UNIVERSITY SERIES IN

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A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher. Problems in Set Theory, Mathematical Logic and the Theory of Algorithms

Igor Lavrov Russian Academy of Sciences Moscow, Russia

and Larisa Maksimova Sobolev Institute of Mathematics of Siberian Branch of Russian Academy of Sciences and Novosibirsk State University Novosibirsk, Russia

Edited by Giovanna Corsi University of Bologna Bologna, Italy

Translated by Valentin Shehtman

Springer Science+Business Media, LLC ISBN 978-1-4613-4957-0 ISBN 978-1-4615-0185-5 (eBook) DOI 10.1007/978-1-4615-0185-5 ©2003 Springer Science+Business Media New York Originally published by Kluwer Academic / Plenum Publishers in 2003 Softcover reprint of tbe hardcover 1st edition 2003 http://www.wkap.com W 9 8 7 6 5 4 3 2 1 The translation of tbis work has been funded by iI* ..

Via Val d' Aposa 7 - 40123 Bologna - ltaly Tel.: +39051 271992 Fax: +39051265983 e-mail: [email protected] - www.seps.it

A C.I.P. record for tbis book is available from the of Congress All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher, witb the exception of any material supplied specifically for tbe purpose of being entered and executed on a computer system, for exclusive use by tbe purchaser of the work Editor's note

This volume owes its origin to the need, felt by every beginner anxious to acquire knowledge and techniques in basic areas of logic, for some sort of gymnasium, as it were, in which (s)he can take exercise and register the results achieved. The judgement of a number of friends and colleagues, whom I warmly thank, convinced me that an English translation of the exercise book" Prob• lems in Set Theory, Mathematical Logic and Theory of Algorithms" by LLavrov and L.Maksimova, well established in Russian Universities, would fill such a gap. Gregory Mints and Sergei Artemov reinforced this conviction and assisted me in the search for a publisher. A final decisive factor was the financial and other assistance provided by the Segretariato Europeo per Ie Pubblicazioni Scientifiche (Bologna). I am especially grateful to its Pres• ident, Professor Fabio Roversi-Monaco, who gave valuable support to the project and to its Secretary General, Chiara Segafredo, who arranged for its approval. My acquaintance with the Segretariato lowe to Guido Gherardi, formerly a student of logic in my own courses.

Italy, August 2002 Giovanna Corsi Department of University of Bologna

v Preface

This book is intended for the student in mathematics, or philosophy. Its aim is to present fundamental theoretic results in logic and to make it possible for the reader to master basic concepts and methods in this foundational area. This is the first presentation in English of the book, which is widely used in Russian universities. In our country, the teaching of the main courses in mathematics is usually organized in the following way. The lectures are followed by training in the subject. The student has to solve a number of exercises in order to learn the basic notions of the subject and to acquire the desirable skills. The contents of such courses and their profit depends mainly on good coordination in the work of the professor and his assistants. Their work must be in parallel. Exercises must be chosen so that the related lecture course will be most effective. But this is not so easy. First of all, there are a lot of useful and necessary exercises but they are scattered in various books. Secondly, each book has its own approach and notation. In our book, we propose a systematic exposition of the first course in mathematical logic and related areas in the form of problems. The book is divided into two parts, each consisting of three chapters: Set Theory, Mathematical Logic and the Theory of Algorithms. Part I consists of exercises, and Part II contains instructions, hints and solutions to the problems of Part I. Each of the chapters is divided into sections, and every section begins with a short introduction containing all the and notations that occur in the section. Notions introduced in preceding sections may be found via the Index. The main are formulated in the form of problems. In order to simplify the proofs, we present the technical lemmas in the same form and give them in separate exercises. Most of the exercises are followed by solu• tions or hints, which are given in the second part of the book. Sometimes we expound a detailed solution of an easy problem in order to illustrate

vii viii PREFACE a method of solving. In subsequent cases we give only a hint. The most difficult exercises are distinguished by an asterisk *. For convenience, the sections of Part II have the same titles and numbers as the sections contain• ing the corresponding problems. In each chapter, many problems may be solved independently of other chapters. When this is not so, are given either in the exercise or in the hints for solution. This structure of the book turned out to be rather convenient. The au• thors hope, with some confidence, that the book is self-contained. The reader may use it for studying Set Theory, Mathematical Logic and Theory (at least the foundations of these) without preliminary preparation. The material presented in our book was extracted from various books on logic, which are included in the References. It is natural that we could not avoid many specialized areas of the theo• ries mentioned, which could not be explored in depth. Some of these topics are only outlined, some initial notions and results are given. For example, axiomatic set theory (see Section 2.7) does not take up much space although all the problems of Chapter 1 may be formulated and solved in the theory ZF. In Chapter 2, devoted to Mathematical Logic, non-classical logical systems are not presented except for intuitionistic logic. In Chapter 3, of the various formalizations of algorithms only recursive functions and Turing machines are presented. The authors planned but, unfortunately, have not written a chapter devoted to the famous G6del incompleteness . Notations and abbreviations usual in mathematical writings are some• times used without explanation. The following may be particularly noted: Z,N, Q, R denote the sets of all , non-negative integers, rational and real numbers respectively; {x I ... x ... } the set of all x satisfying the condition ... x ... ; {Xl, X2, ••. } the set consisting of Xl. X2, •• ·; (x!, X2, ••• , xn ) the sequence of elements Xl, X2, ••• , Xn ; =} stands for "implies"; {::} for ""; ;:::= for" is by " .

The first edition of our book in Russia arose from our teaching at Novosi• birsk State University. Now the book has been widely used in Russian universities. In 2001 its fourth edition was published. There are also trans• lations into Hungarian and Polish. In every new edition we tried to take into account all suggestions and remarks of our colleagues actively using the book in their teaching or research. To all these persons we express our deep gratitude. ix

The present edition has involved both translation and revision. Special thanks are due to Valentin Shehtman, who prepared the translation into En• glish and suggested many improvements, and to Aleksey Romanov and Ilya Shapirovsky, who typed the formulas in LATEX. We are greatly indebted to Giovanna Corsi, our editor of the translation, whose initiative and activity made possible the publication of the English version of the book, and also to the Segretariato Europeo per Ie Pubblicazioni Scientifiche (SEPS), which supported the project.

Russia, August 2002 Igor Lavrov Larisa Maksimova Department of Mathematics Novosibirsk State University Russian Academy of Sciences and Moscow Sobolev Institute of Mathematics Siberian Branch of Russian Academy of Sciences Contents

Preface vii

I Problems 1

1 Set theory 3 1.1 Operations on sets 3 1.2 Relations and functions 11 1.3 Special binary relations 19 1.4 Cardinal numbers. . . . 30 1.5 Ordinal numbers . . . . 34 1.6 Operations on cardinal numbers. 44

2 Mathematical logic 51 2.1 of 51 2.2 functions ..... 60 2.3 Propositional calculi .. 67 2.4 The of predicate logic 79 2.5 of predicate formulas 88 2.6 Predicate calculi . 96 2.7 Axiomatic . . 107 2.8 Reduced products .. 118 2.9 Axiomatizable classes 126

3 Theory of algorithms 135 3.1 Partial recursive functions 135 3.2 Turing machines ..... 148 3.3 Recursive and recursively enumerable sets 154 3.4 Kleene and Post numberings...... 160

xi xii CONTENTS

II Solutions 167

1 Set theory 169 1.1 Operations on sets 169 1.2 Relations and functions 174 1.3 Special binary relations 179 1.4 Cardinal numbers. . . . 185 1.5 Ordinal numbers . . . . 190 1.6 Operations on cardinal numbers. 198

2 Mathematical logic 203 2.1 Algebra of propositions 203 2.2 Truth functions ..... 207 2.3 Propositional calculi . . 212 2.4 The language of predicate logic 217 2.5 Satisfiability of predicate formulas 219 2.6 Predicate calculi . 225 2.7 Axiomatic theories . . 228 2.8 Reduced products .. 235 2.9 Axiomatizable classes 239

3 Theory of algorithms 249 3.1 Partial recursive functions ...... 249 3.2 Turing machines ...... 256 3.3 Recursive and recursively enumerable sets 260 3.4 Kleene and Post numberings...... 265

References 271

Index 275