Texts in Computing - SQIG

19 Pages·2008·1.22 MB·English

Texts in
Computing
Volume 10
Foundations of Logic and
Theory of Computation
Volume 3
Logical Reasoning: A First Course
Rob Nederpelt and Fairouz Kamareddine
Volume 4
The Haskell Road to Logic, Maths and Programming
Kees Doets and Jan van Eijck
Volume 5
Bridges from Classical to Nonmonotonic Reasoning
David Makinson
Volume 6
Automata and Dictionaries
Denis Maurel and Franz Guenthner
Volume 7
Learn Prolog Now!
Patrick Blackburn, Johan Bos and Kristina Striegnitz
Volume 8
A Meeting of the Minds: Proceedings of the Workshop on Logic,
Rationality and Interaction, Beijing, 2007
Johan van Benthem, Shier Ju and Frank Veltman, eds.
Volume 9
Logic for Artificial Intelligence & Information Technology
Dov M. Gabbay
Volume 10
Foundations of Logic and Theory of Computation
Am´ılcar Sernadas and Cristina Sernadas
Texts in Computing Series Editor
Ian Mackie [email protected]
Foundations of Logic
and
Theory of Computation
Am(cid:19)(cid:16)lcar Sernadas and Cristina Sernadas
(cid:13)c Individual author and College Publications, 2008. All rights reserved.
ISBN 978-1-904987-88-8
College Publications
Scientific Director: Dov Gabbay
Managing Director: Jane Spurr
Department of Computer Science
King’s College London
Strand, London WC2R 2LS, UK
Original cover design by Richard Fraser
Cover produced by orchid creative www.orchidcreative.co.uk
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
ortransmitted,inanyform,orbyanymeans,electronic,mechanical,photocopying,recordingor
otherwise,withoutpriorpermission,inwriting,fromthepublisher.
Preface
Themainobjectiveofthebookistoprovideaself-containedintroductionto
mathematical logic and computability theory for students of Mathematics
or Computer Science. It starts with the basics of computability theory,
presents the language, semantics, Hilbert calculus and Gentzen calculus
of first-order logic, develops the notion of first-order theory, proves the
incompleteness theorems about arithmetic and concludes with a decidable
theory of arithmetic.
The material is organized around the failures and successes of Hilbert’s
programme for the formalization of Mathematics. It is widely known that
the programme failed with G¨odel’s incompleteness theorems and related
negative results about arithmetic. Unfortunately, the positive outcomes of
theprogrammearelesswellknown,evenamongmathematicians. Thebook
covers key successes, like G¨odel’s proof of the completeness of first-order
logic, Gentzen’s proof of its consistency by purely symbolic means, and
the decidability of a couple of useful theories, including Presburger arith-
metic. It also tries to convey the message that Hilbert’s programme made
a significant contribution to the advent of the computer as it is nowadays
understood and, thus, to the latest industrial revolution.
Part I starts with Hilbert’s programme and moves on to computability.
Part II presents first-order logic, including G¨odel’s completeness theorem
and Gentzen’s consistency theorem. Part III is focused on arithmetic, rep-
resentability of computable maps, G¨odel’s incompleteness theorems and
decidability of Presburger arithmetic. Part IV provides detailed answers to
selected exercises.
Thebookcanbeusedinseveralwaysatlateundergraduatelevelorearly
graduate level. An undergraduate course would concentrate on Parts I and
II, leaving out Chapter 9, and sketching the way to the 1st incompleteness
v
vi
theorem. A more advancedcoursemightskip early materialalready known
to the students and concentrate on the positive and negative results of
Hilbert’s programme, thus covering Chapter 9 and Part III in full. Other
paths through the book are possible as depicted in Figure 1.
The text is an outgrowth of the translation1 into English of the 2006
Portuguese draft. It is the result of many years of experience of teaching
logic and computability to undergraduate and graduate students.
The authors would like to express their deepest gratitude to their stu-
dents who reacted to early drafts of the book, several colleagues (especially
Lu´ısCruz-Filipe,PauloMateus,JaimeRamosandJo˜aoRasga)whohelped
in the debugging stage, and the anonymous referee for many suggestions.
The remaining mistakes and omissions are of course the sole responsibility
of the authors.
The excellent working environment provided by the Department of
Mathematics of Instituto Superior T´ecnico (IST) and the Security and
Quantum Information Group (SQIG) of Instituto de Telecomunica¸c˜oes is
also acknowledged by the authors.
Lisbon, March 14, 2008.
Am´ılcar Sernadas
Cristina Sernadas
Department of Mathematics
Instituto Superior T´ecnico
Universidade T´ecnica de Lisboa
Security and Quantum Information Group
Instituto de Telecomunicac¸˜oes
1By our colleague Lu´ıs Cruz-Filipe.
vii
1. Preliminaries
(cid:178)(cid:178)
2. Hilbert’s programme
(cid:178)(cid:178)
3. Computability
(cid:178)(cid:178)
4. FOL syntax
(cid:178)(cid:178)
5. FOL axiomatization
(cid:178)(cid:178)
6. FOL theories
(cid:178)(cid:178)
7. FOL semantics
(cid:117)(cid:117)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108) (cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:41)(cid:41)
8. FOL completeness 9. FOL sequent calculus
(cid:178)(cid:178)
10. Equality
(cid:178)(cid:178)
11. Arithmetic
(cid:117)(cid:117)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108)(cid:108) (cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:82)(cid:41)(cid:41)
12. Representability 15. Decidable arithmetic
(cid:178)(cid:178)
13. 1st incompleteness theorem
(cid:178)(cid:178)
14. 2nd incompleteness theorem
Figure 1: Chapter dependence graph
viii
Contents
1 Preliminaries 1
1.1 Functions versus maps . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lambda notation . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Inductive definitions as fixed points . . . . . . . . . . . . . . 2
1.4 Proofs by induction. . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Programming in Mathematica . . . . . . . . . . . . . . . . . 7
I Formalizing Mathematics 11
2 Hilbert’s Programme 13
3 Computability theory 17
3.1 Formal languages . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Church–Turing postulate. . . . . . . . . . . . . . . . . . . . 19
3.3 Computability in Mathematica . . . . . . . . . . . . . . . . 20
3.4 G¨odelizations . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Computability `a la Kleene . . . . . . . . . . . . . . . . . . . 36
3.6 Undecidability of the halting problem . . . . . . . . . . . . 40
3.7 Enumerating the computable functions . . . . . . . . . . . . 42
II First-order predicate logic 45
4 Syntax 47
4.1 Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Terms and formulas . . . . . . . . . . . . . . . . . . . . . . 50
ix
x CONTENTS
4.3 Free and bound variables . . . . . . . . . . . . . . . . . . . 54
4.4 Free terms and substitutions . . . . . . . . . . . . . . . . . 56
5 Hilbert calculus 61
5.1 Axioms and rules . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Derivation as a fixed point . . . . . . . . . . . . . . . . . . . 65
5.3 Properties of derivation . . . . . . . . . . . . . . . . . . . . 69
5.4 Metatheorems and admissible rules . . . . . . . . . . . . . . 76
6 Theories and presentations 91
6.1 Formalizing mathematical concepts . . . . . . . . . . . . . . 91
6.2 Decidability and semi-decidability . . . . . . . . . . . . . . 94
6.3 Quantifier elimination . . . . . . . . . . . . . . . . . . . . . 99
7 Semantics 105
7.1 Interpretation structures . . . . . . . . . . . . . . . . . . . . 106
7.2 Satisfaction, entailment and validity . . . . . . . . . . . . . 108
7.3 Basic lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4 Soundness of the Hilbert calculus . . . . . . . . . . . . . . . 119
7.5 Theory of interpretation structure . . . . . . . . . . . . . . 123
7.6 Relating interpretation structures . . . . . . . . . . . . . . . 123
8 Completeness 131
8.1 Lindenbaum’s lemma . . . . . . . . . . . . . . . . . . . . . . 132
8.2 Henkin construction . . . . . . . . . . . . . . . . . . . . . . 135
8.3 G¨odel’s completeness theorem . . . . . . . . . . . . . . . . . 144
9 Gentzen calculus 149
9.1 Rules and derivations . . . . . . . . . . . . . . . . . . . . . 151
9.2 Cut elimination and consistency . . . . . . . . . . . . . . . 157
9.3 Consistency of first-order logic . . . . . . . . . . . . . . . . 167
9.4 Soundness and completeness . . . . . . . . . . . . . . . . . . 171
10 Equality 177
10.1 Theories with equality . . . . . . . . . . . . . . . . . . . . . 177
10.2 Normal models . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.3 First-order logic with equality . . . . . . . . . . . . . . . . . 187