An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof

403 Pages·2002·12.793 MB·English

by  Peter B. Andrews (auth.)

An Introduction to Mathematical Logic and Type Theory:
To Truth Through Proof
APPLIED LOGIC SERIES
VOLUME27
Managing Editor
Dov M. Gabbay, Department of Computer Science, King's College, London,
U.K.
Co-Editor
Jon Barwiset
Editorial Assistant
Jane Spurr, Department of Computer Science, King's College, London, U.K.
SCOPE OF THE SERIES
Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects
of philosophy and mathematics to the more recent disciplines of cognitive science, compu
ter science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject.
Kluwer, through its Applied Logic Series, seeks to provide a home for outstanding books and
research monographs in applied logic, and in doing so demonstrates the underlying unity and
applicability of logic.
An Introduction to
Mathematical Logic and
Type Theory:
To Truth Through Proof
Second Edition
by
PETER B. ANDREWS
Department of Mathematical Sciences,
Carnegie Mellon University,
Pittsburgh, Pennsylvania, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6079-2 ISBN 978-94-015-9934-4 (eBook)
DOI 10.1007/978-94-015-9934-4
Printed an acid-free paper
AlI Rights Reserved
© Peter B. Andrews 2002
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 2nd edition 2002
No part of this work 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, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
To Gate, Lyle, and Bruce
Contents
Preface to the Second Edition ix
Preface xi
0 Introduction 1
1 Propositional Calculus 5
§1 0. The Language of 'P . . . . . . . 5
§lOASupplement on Induction ... 17
§11. The Axiomatic Structure of 'P . 21
§12. Semantics, Consistency,
and Completeness of 'P . . 32
§13. Independence ...... . 42
§14. Propositional Connectives 44
§15. Compactness . . . 64
§16. Ground Resolution 67
2 First-Order Logic 73
§20. The Language of :F ...... . 73
§21. The Axiomatic Structure of :F . 84
§22. Prenex Normal Form . 110
§23. Semantics of :F . . . . 114
§24. Independence . . . . . 123
§25. Abstract Consistency
and Completeness . . 125
§25ASupplement:
Simplified Completeness Proof 139
§26. Equality . . . . . . . . . . . 142
3 Provability and Refutability 151
§30. Natural Deduction 151
§31. Gentzen's Theorem . 156
§32. Semantic Tableaux . 160
§33. Skolemization . . . . 166
§34. Refutations of Universal Sentences 171
§35. Herbrand's Theorem 176
§36. Unification . . . . . . . . 183
vii
viii
4 Further Topics in First-Order Logic 189
§40. Duality . . . . . . . . . . . . . 189
§41. Craig's Interpolation Theorem 196
§42. Beth's Definability Theorem . 199
5 Type Theory 201
§50. Introduction . ....... 201
§51. The Primitive Basis of Qo 210
§52. Elementary Logic in Qo 215
§53. Equality and Descriptions 232
§54. Semantics of Qo .. 238
§55. Completeness of Qo ... 248
6 Formalized Number Theory 257
§60. Cardinal Numbers and
the Axiom of Infinity . 257
§61. Peano's Postulates 261
§62. Order ....... . 269
§63. Minimization . . . . 276
§64. Recursive Functions 281
§65. Primitive Recursive Functions
and Relations . . . . . . . . . . 289
7 Incompleteness and Undecidability 301
§70. Godel Numbering ........ . 303
§71. Godel's Incompleteness Theorems . 312
§72. Essential Incompleteness ..... . 323
§73. Undecidability and Undefinability. 332
§74. Epilogue .............. . 338
Supplementary Exercises 339
Summary of Theorems 345
Bibliography 371
List of Figures 381
Index 383
lX
Preface to the Second Edition
For the benefit of readers who are already familiar with the first (1986)
edition of this book, we briefly explain here how this edition differs from
that one.
The basic structure of the book is unchanged, but improvements and
enhancements may be found throughout the text. There is more introduc
tory material and motivation. Historical notes and bibliographic references
have been added. The discussion of vertical paths and normal forms in §14
is now much more comprehensive. A number of consequences of Godel's
Second Incompleteness Theorem are now discussed in §7 1.
Many exercises have been added, and at the end of the book there is
a new collection of Supplementary Exercises which are not explicitly tied
to any particular section. Information about obtaining the ETPS program,
which facilitates work on exercises which consist of proving theorems in the
object language, will be found near the end of the Preface.
Preface
This book is an introductory text on mathematical logic and type theory.
It is aimed primarily at providing an introduction to logic for students of
mathematics, computer science, or philosophy who are at the college junior,
senior, or introductory graduate level. It can also be used as an introduction
to type theory by researchers at more advanced levels.
The first part of the book (Chapters 1 and 2, supplemented by parts of
Chapters 3 and 4) is suitable for use as a text in a one-semester introduction
to mathematical logic, and the second part (Chapters 5 - 7) for a second
semester course which introduces type theory (more specifically, typed A
calculus) as a language for formalizing mathematics and proves the classical
incompleteness and undecidability theorems in this context. Persons who
wish to learn about type theory and have had a prior introduction to logic
will have no difficulty starting with Chapter 5.
The book is oriented toward persons who wish to study logic as a ve
hicle for formal reasoning. It may be particularly useful to those who are
interested in issues related to the problem of constructing formal proofs as
models of informal reasoning or for use in computerized systems which in
volve automated deduction. Proofs, which are often the chief end products
and principal manifestations of mathematical reasoning, constitute highly
significant pathways to truth and are a central concern of this book. Our
choice of the title To Truth Through Proof is motivated by the consideration
that while in most realms one needs more than logic to achieve an under
standing of what is true, in mathematics the primary and ultimate tool for
establishing truth is logic.
Of course, the study of logic involves reasoning about reasoning, and it
is not surprising that complex questions arise. To achieve deep understand
ing and proper perspective, one must study a variety of logical systems as
mathematical objects and look at them from a variety of points of view. We
are thus led to study the interplay between syntax and semantics, questions
of consistency and independence, formal rules of reasoning, various formats
for proofs and refutations, ways of representing basic mathematical concepts
in a formal system, the notion of computability, and the completeness, in
completeness, and undecidability theorems which illuminate both the power
and the limitations of logic.
One of the basic tasks of mathematical logic is the formalization of math
ematical reasoning. Both type theory (otherwise known as higher-order
logic) and axiomatic set theory can be used as formal languages for this
xu PREFACE
purpose, and it is really an accident of intellectual history that at present
most logicians and mathematicians are more familiar with axiomatic set
theory than with type theory. It is hoped that this book will help to remedy
this situation.
In logic as in other realms, there is a natural tendency for people to prefer
that with which they are most familiar. However, those familiar with both
type theory and axiomatic set theory recognize that in some ways the former
provides a more natural vehicle than the latter for formalizing what mathe
maticians actually do. Both logical systems are necessarily more restrictive
than naive axiomatic set theory with the unrestricted Comprehension Ax
iom, since the latter theory is inconsistent. Axiomatic set theory achieves
consistency by restricting the Comprehension Axiom and introducing the
distinction between sets and classes. Mathematicians often find this distinc
tion unnatural, ignore the technicalities about existence axioms, and leave
it to the specialists to show that their reasoning can be justified. Type the
ory achieves consistency by distinguishing between different types of objects
(such as numbers, sets of numbers, collections of sets of numbers, functions
from numbers to numbers, sets of such functions, etc.). Mathematicians
make such distinctions too, and even use different letters or alphabets to
help them distinguish between different types of objects, so the restrictions
which type theory introduces are already implicit in mathematical practice.
While some formulations of type theory may seem cumbersome, the formu
lation Qo introduced in Chapter 5 is a very rich and expressive language,
with functions (which need not be regarded as sets of ordered pairs) of all
types as primitive objects, so most of what mathematicians write can be
translated into Qo very directly. Qo is a version of typed A-calculus, and
the availability of A-notation in Qo enables definitions to be handled very
conveniently and eliminates the need for axioms asserting the existence of
sets and functions.
One's choice of a formal language will generally depend on what one
wishes to do with it. If one is choosing a language for expressing mathematics
in computerized systems which deal with mathematics on a theoretical level
(such as mathematically sophisticated information retrieval systems, proof
checkers, or deductive aids), there are several reasons why type theory may
be preferable to set theory. First, the primitive notation of set theory is so
economical that this language is only practical to actually use if one expands
it by introducing various abbreviations. This is usually done in an informal
way (i.e., in the meta-language), but in an automated system it is important
that the formal language be the one actually used to express mathematical
statements, since this is the language which must be studied rigorously and