A Beginner's Further Guide to Mathematical Logic

288 Pages·2017·4.978 MB·English

by  Raymond Smullyan

b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Smullyan, Raymond M., author.
Title: A beginner’s further guide to mathematical logic / Raymond Smullyan.
Description: New Jersey : World Scientific, 2016. |
Includes bibliographical references and index.
Identifiers: LCCN 2015033651 | ISBN 9789814730990 (hardcover : alk. paper) |
ISBN 9789814725729 (pbk : alk. paper)
Subjects: LCSH: Logic, Symbolic and mathematical.
Classification: LCC QA9.A1 S619 2016 | DDC 511.3--dc23
LC record available at http://lccn.loc.gov/2015033651
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
On the cover, the three photos from left to right are the logicians
Emil Post, Alan Turing, and Ernst Zermelo.
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Contents
Preface ix
Part I: More on Propositional and First-Order Logic 1
Chapter 1. More on Propositional Logic 3
I. Propositional Logic and the Boolean Algebra of Sets . . . 3
II. An Algebraic Approach . . . . . . . . . . . . . . . . . . . 4
III. Another Completeness Proof . . . . . . . . . . . . . . . . 7
IV. Fidelity to Modus Ponens . . . . . . . . . . . . . . . . . . 13
Chapter 2. More on First-Order Logic 23
I. Magic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 23
II. Gentzen Sequents and Some Variants. . . . . . . . . . . . 29
III. Craig’s Lemma and an Application . . . . . . . . . . . . . 43
IV. A Unification . . . . . . . . . . . . . . . . . . . . . . . . . 48
V. A Henkin-Style Completeness Proof . . . . . . . . . . . . 54
v
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vi Contents
Part II: Recursion Theory and Metamathematics 67
Chapter 3. Some Special Topics 69
I. A Decision Machine . . . . . . . . . . . . . . . . . . . . . 69
II. Variations on a Theme of G¨odel. . . . . . . . . . . . . . . 73
III. R-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
IV. A Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Chapter 4. Elementary Formal Systems and Recursive
Enumerability 89
I. More on Elementary Formal Systems . . . . . . . . . . . . 89
II. Recursive Enumerability . . . . . . . . . . . . . . . . . . . 93
III. A Universal System . . . . . . . . . . . . . . . . . . . . . 99
Chapter 5. Some Recursion Theory 113
I. Enumeration and Iteration Theorems. . . . . . . . . . . . 113
II. Recursion Theorems . . . . . . . . . . . . . . . . . . . . . 117
Chapter 6. Doubling Up 133
Chapter 7. Metamathematical Applications 149
I. Simple Systems . . . . . . . . . . . . . . . . . . . . . . . . 149
II. Standard Simple Systems . . . . . . . . . . . . . . . . . . 154
Part III: Elements of Combinatory Logic 171
Chapter 8. Beginning Combinatory Logic 173
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Contents vii
Chapter 9. Combinatorics Galore 185
I. The B-Combinators . . . . . . . . . . . . . . . . . . . . . 185
II. The Permuting Combinators . . . . . . . . . . . . . . . . 187
III. The Q-Family and the Goldfinch, G . . . . . . . . . . . . 190
IV. Combinators Derivable from B,T,M and I
(λ-I Combinators) . . . . . . . . . . . . . . . . . . . . . . 192
Chapter 10. Sages, Oracles and Doublets 205
Chapter 11. Complete and Partial Systems 215
I. The Complete System . . . . . . . . . . . . . . . . . . . . 215
II. Partial Systems of Combinatory Logic . . . . . . . . . . . 224
Chapter 12. Combinators, Recursion and the Undecidable 233
I. Preparation for the Finale . . . . . . . . . . . . . . . . . . 236
II. The Grand Problem . . . . . . . . . . . . . . . . . . . . . 239
Afterword. Where to Go from Here 253
References 261
Index 265
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Preface
This book is a sequel to my Beginner’s Guide to Mathematical Logic
[Smullyan, 2014]. I originally intended both volumes to be a single vol-
ume, but I felt that at my age (now 96), I could pass away at any time,
and I wanted to be sure that I would at least get the basic material out.
Thepreviousvolumedealswithelementsofpropositionalandfirst-order
logic, contains a bit on formal systems and recursion, and concludes with
chapters on G¨odel’s famous incompleteness theorem, along with related
results.
The present volume begins with a bit more on propositional and first-
orderlogic,followedbywhatIwouldcalla“fein”chapter,whichsimultane-
ously generalizes some results from recursion theory, first-order arithmetic
systems, and what I dub a “decision machine.” Then come four chapters
on formal systems, recursion theory and metamathematical applications in
a general setting. The concluding five chapters are on the beautiful subject
of combinatory logic, which is not only intriguing in its own right, but has
important applications to computer science. Argonne National Laboratory
is especially involved in these applications, and I am proud to say that its
members have found use for some of my results in combinatory logic.
This book does not cover such important subjects as set theory, model
theory,prooftheory,andmoderndevelopmentsinrecursiontheory,butthe
reader, after studying this volume, will be amply prepared for the study of
these more advanced topics.
Although this book is written for beginners, there are two chapters −
namely 3 and 8 − that I believe would also be of interest to the expert.
For brevity, all references to the first volume, The Beginner’s Guide to
Mathematical Logic, of this two-volume introduction to mathematical logic
will be given in the remainder of this volume as The Beginner’s Guide
[Smullyan,2014].
Elka Park
November 2016
ix