Table Of ContentBeginning Mathematical Logic: A Study Guide
There are many wonderful introductory texts on mathematical
logic, but there are also many not-so-useful books. So how do you
find your way around the very large literature old and new, and
how do you choose what to read? Beginning Mathematical Logic
provides the necessary guide. It introduces the core topics and
recommends the best books for studying these topics enjoyably
and effectively. This will be an invaluable resource both for those
wanting to teach themselves new areas of logic and for those
looking for supplementary reading before or during a university
course.
Peter Smith was formerly Senior Lecturer in Philosophy at the
University of Cambridge, and taught logic for more years than he
cares to remember. His books include Explaining Chaos (1998),
An Introduction to Formal Logic (2003; 2020), An Introduction to
G¨odel’s Theorems (2007; 2013), and G¨odel Without (Too Many)
Tears (2020). He was also editor of Analysis for a dozen years.
The page intentionally left blank
Beginning Mathematical Logic
A Study Guide
Peter Smith
LOGIC MATTERS
Published by Logic Matters, Cambridge
© Peter Smith 2022
All rights reserved. Permission is granted to privately share this PDF
as a complete whole, including this copyright page, for educational
purposes. But it is not to be hosted on a website or otherwise
distributed in whole or in part, without prior written permission. For
permission requests, write to peter [email protected].
A paperback copy of this book is available by print-on-demand from
Amazon: ISBN 978-1-91690633-4.
Additional resources are available at www.logicmatters.net/tyl where
the latest version of this PDF will always be found.
This version: February 18, 2022.
Since this book is not produced by a publisher with a marketing
department, your university librarian will not get to know about it in
the usual way. You will therefore need to ask them to order a printed
copy for the library. Please do so, if you find this book useful.
Contents
Preface ix
1 The Guide, and how to use it 1
1.1 Who is the Guide for? 1
1.2 The Guide’s structure 1
1.3 Strategies for self-teaching from logic books 4
1.4 Choices, choices 5
1.5 So what do you need to bring to the party? 6
1.6 Two notational conventions 7
2 A very little informal set theory 8
2.1 Sets: a checklist of some basics 8
2.2 A note about naivety 10
2.3 Recommendations on informal basic set theory 10
2.4 Virtual classes, real sets 11
3 First-order logic 13
3.1 Propositional logic 13
3.2 FOL basics 16
3.3 A little more about types of proof-system 20
3.4 Basic recommendations for reading on FOL 24
3.5 Some parallel and slightly more advanced reading 26
3.6 A little history (and some philosophy too) 30
3.7 Postscript: Other treatments? 32
4 Second-order logic, quite briefly 37
4.1 A preliminary note on many-sorted logic 37
4.2 Second-order logic 39
4.3 Recommendations on many-sorted and second-order logic 44
4.4 Conceptual issues 45
5 Model theory 47
5.1 Elementary model theory 47
5.2 Recommendations for beginning first-order model theory 53
5.3 Some parallel and slightly more advanced reading 55
5.4 A little history 57
6 Arithmetic, computability, and incompleteness 59
6.1 Logic and computability 59
v
Contents
6.2 Computable functions 61
6.3 Formal arithmetic 64
6.4 Towards Go¨delian incompleteness 66
6.5 Main recommendations on arithmetic, etc. 67
6.6 Some parallel/additional reading 70
6.7 A little history 73
7 Set theory, less naively 74
7.1 Set theory and number systems 74
7.2 Ordinals, cardinals, and more 78
7.3 Main recommendations on set theory 83
7.4 Some parallel/additional reading on standard ZFC 85
7.5 Further conceptual reflection on set theories 88
7.6 A little more history 88
7.7 Postscript: Other treatments? 89
8 Intuitionistic logic 90
8.1 A formal system 90
8.2 Why intuitionistic logic? 91
8.3 More proof theory, more semantics 94
8.4 Basic recommendations on intuitionistic logic 98
8.5 Some parallel/additional reading 99
8.6 A little more history, a little more philosophy 100
9 Elementary proof theory 102
9.1 Preamble: a very little about Hilbert’s Programme 102
9.2 Deductive systems, normal forms, and cuts 103
9.3 Proof theory and the consistency of arithmetic 111
9.4 Main recommendations on elementary proof theory 114
9.5 Some parallel/additional reading 117
10 Modal logics 120
10.1 Some basic modal logics 120
10.2 Provability logic 127
10.3 First readings on modal logic 131
10.4 Suggested readings on provability logic 133
10.5 Alternative and further readings on modal logics 134
10.6 Finally, a very little history 135
11 Other logics? 137
11.1 Relevant logic 137
11.2 Readings on relevant logic 143
11.3 Free logic 144
11.4 Readings on free logic 151
11.5 Plural logic 152
11.6 Readings on plural logic 157
12 Going further 159
12.1 A very little light algebra for logic? 160
12.2 More model theory 161
vi
Contents
12.3 More on formal arithmetic and computability 164
12.4 More on mainstream set theory 169
12.5 Choice, and the choice of set theory 172
12.6 More proof theory 177
12.7 Higher-order logic, the lambda calculus, and type theory 178
Index of authors 182
vii
The page intentionally left blank
Preface
Thisisnotanothertextbookonmathematicallogic:itisaStudyGuide,abook
mostly about textbooks on mathematical logic. Its purpose is to enable you to
locate the best resources for teaching yourself various areas of logic, at a fairly
introductory level. Inevitably, given the breadth of its coverage, the Guide is
rather long: but don’t let that scare you off! There is a good deal of signposting
andtherearealsoexplanatoryoverviewstoenableyoutopickyourwaythrough
and choose the parts which are most relevant to you.
Beginning Mathematical Logic isadescendantofmymuch-downloadedTeach
Yourself Logic. The new title highlights that the Guide focuses mainly on the
core mathematical logic curriculum. It also signals that I do not try to cover
advanced material in any detail.
ThefirstchaptersaysmoreaboutwhotheGuideisintendedfor,whatitcovers,
and how to use it. But let me note straightaway that most of the main reading
recommendations doindeed pointtopublished books. True, there arequite alot
of on-line lecture-notes that university teachers have made available. Some of
these are excellent. However, they do tend to be terse, and often very terse (as
entirely befits material originally intended to support a lecture course). They
are therefore usually not as helpful as fully-worked-out book-length treatments,
at least for students needing to teach themselves.
So where can you find the titles mentioned here? I suppose I ought to pass
overtheissueofdownloadingbooksfromcertainverywell-knownandextremely
well-stockedcopyright-infringingPDFrepositories.That’sbetweenyouandyour
conscience (though almost all the books are available to be sampled there).
Anyway,manydoprefertoworkfromphysicalbooks.Mostofthesetitlesshould
infactbeheldbyanylarge-enoughuniversitylibrarywhichhasbeentryingover
theyearstomaintaincorecollectionsinmathematicsandphilosophy(andifthe
locallibraryistoosmall,booksshouldbeborrowablethroughsomeinter-library
loans system).
Since I’m not assuming that you will be buying the recommended books,
I have not made cost or being currently in print a significant consideration.
However, I have marked with a star* books that are available new or second-
handrelativelyinexpensively(oratleastareunusuallygoodvalueforthelength
and/or importance of the book). When e-copies of books are freely and legally
available,linksareprovided.Wherejournalarticlesorencyclopaediaentrieshave
ix
Preface
been recommended, these can almost always be freely downloaded, and again I
give links.
Before I retired from the University of Cambridge, it was my greatest good
fortunetohavesecure,decentlypaid,universitypostsforfortyyearsinleisurely
times,withalmosttotalfreedomtofollowmyinterestswherevertheymeandered.
Likemostofmycontemporaries,formuchofthattimeIdidn’treallyappreciate
how extraordinarily lucky I was. In writing this Study Guide and making it
readily available, I am trying to give a little back by way of heartfelt thanks. I
hope you find it useful.1
1Iowemuchtothekindnessofstrangers:manythanks,then,toallthosewhocommentedon
earlierversionsofTeach Yourself Logic andBeginning Mathematical Logic overadecade,
far too many to list here. I am particularly grateful though to Rowsety Moid for all his
suggestionsovertheyears,andforalengthysetofcommentswhichledtomanylast-minute
improvements.
FurthercommentsandsuggestionsforapossiblerevisededitionofthisGuidewillalways
bemostwlecome.
Athena’s familiar at the very end of the book is borrowed from the final index page
of the 1794 Clarendon Press edition of Aristotle’s Poetics, with thanks to McNaughtan’s
Bookshop,Edinburgh.
x
