Set theory and its philosophy: a critical introduction

360 Pages·2004·1.401 MB·English

by  Michael Potter

Set Theory and its Philosophy
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Set Theory and its Philosophy
A Critical Introduction
Michael Potter
3
3
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Preface
This book was written for two groups of people, philosophically informed
mathematicianswithaseriousinterestinthephilosophyoftheirsubject,and
philosophers with a serious interest in mathematics. Perhaps, therefore, I
should begin by paying tribute to my publisher for expressing no nervous-
ness concerning the size of the likely readership. Frege (1893–1903, I, p.xii)
predictedthattheformofhisGrundgesetzewouldcausehimto
relinquishasreadersallthosemathematicianswho,iftheybumpintologicalexpres-
sionssuchas‘concept’,‘relation’,‘judgment’,think: metaphysicasunt,nonleguntur,and
likewise those philosophers who at the sight of a formula cry: mathematica sunt, non
leguntur.
And,asherightlyobserved,‘thenumberofsuchpersonsissurelynotsmall’.
As then, so now. Any book which tries to form a bridge between math-
ematicsandphilosophyrisksvanishingintothegap. Itisinevitable,however
hardthewriterstrivesforclarity,thattherequirementsofthesubjectmatter
placedemandsonthereader,sometimesmathematical,sometimesphilosoph-
ical. This is something which anyone who wants to make a serious study of
thephilosophyofmathematicsmustsimplyaccept. Toanyonewhodoubtsit
therearetwobodiesofworkwhichstandasanawfulwarning: thephilosoph-
ical literature contains far too many articles marred by elementary technical
misunderstandings,1 while mathematicians have often been tempted, espe-
ciallyinlaterlife,tocommittoprintphilosophicalreflectionswhichareeither
whollyvacuousorhopelesslyincoherent.
Bothmathematiciansandphilosophers,then,needtoacceptthatstudying
thephilosophyofmathematicsconfrontsthemwithchallengesforwhichtheir
previous training may not have prepared them. It does not follow automat-
ically, of course, that one should try, as I have done here, to cater for their
differingneedsinasingletextbook. However,mymainreasonforwritingthe
bookwasthatIwantedtoexploretheconstantinterplaythatsettheoryseems
to exemplify between technical results and philosophical reflection, and this
convinced me that there was more than expositional economy to be said for
tryingtoaddressbothreadershipssimultaneously.
1Ishouldknow:Ihavewrittenonemyself.
vi Preface
In this respect the book differs from its predecessor, Sets: An Introduction
(1990), which was directed much more at philosophically ignorant mathem-
aticiansthanatphilosophers. Thesimilaritiesbetweenthenewbookandthe
old are certainly substantial, especially in the more technical parts, but even
therethechangesgowellbeyondthecosmetic. Technicallyinformedreaders
mayfindaquicksummaryhelpful.
At the formal level the most significant change is that I have abandoned
the supposition I made inSets thatthe universe of collections contains a sub-
universe which has all the sets as members but is nevertheless capable of be-
longingtoothercollections(not,ofcourse,sets)itself. ThetruthisthatIonly
includeditfortworeasons: tomakeiteasytoembedcategorytheory;andto
sidestepsomeirritatingpiecesofpedantrysuchasthedifficultywefacewhen
we try, without going metalinguistic, to say in standard set theory that the
classofallordinalsiswell-ordered. Neitherreasonnowstrikesmeassufficient
tocompensate for thecomplications positing a sub-universe creates: the cat-
egorytheoristsneverthankedmeforaccommodatingthem,andiftheywant
a sub-universe, I now think that they can posit it for themselves; while the
pedantryisgoingtohavetobefacedatsomepoint,sopostponingitdoesno
oneanyfavours.
The style of set-theoretic formalism in which the sets form only a sub-
universe (sometimes in the literature named after Grothendieck) has never
found much favour. (Perhaps category theory is not popular enough.) So in
thisrespectthenewbookisthemoreorthodox. Intwootherrespects,though,
Ihavepersistedineccentricity: Istillallowtheretobeindividuals; andIstill
donotincludetheaxiomschemeofreplacementinthedefaulttheory.
The reasons for these choices are discussed quite fully in the text, so here
I will be brief. The first eccentricity — allowing individuals — seems to me
tobesomethingclosetoaphilosophicalnecessity: asitcomplicatesthetreat-
ment only very slightly, I can only recommend that mathematicians who do
notseetheneedshouldregarditasafoibleandhumourmeinit. Thesecond
—doingwithoutreplacement—wasinfactalargepartofmymotivationto
writeSetstobeginwith. Ihadwhiledawaymystudentdaysunderthedelusion
that replacement is needed for the formalization of considerable amounts of
mathematics,andwhenIdiscoveredthatthiswasfalse,Iwantedtospreadthe
word (spread it, that is to say, beyond the set theorists who knew it already).
Onthispointmyproselytizingzealhashardlywanedintheinterveningdec-
ade. OneofthethemeswhichIhavetriedtodevelopinthenewbookisthe
idea that set theory is a measure (not the only one, no doubt) of the degree
ofabstractnessofmathematics,anditisattheveryleastastrikingfactabout
mathematicalpractice,whichmanysettheorytextbookscontrivetoobscure,
that even before we try to reduce levels by clever use of coding, the over-
whelming majority of mathematics sits comfortably inside the first couple of
dozenlevelsofthehierarchyabovethenaturalnumbers.
Preface vii
ThedifferencesfromstandardtreatmentsthatIhavedescribedsofaraffect
our conception of the extent of the set-theoretic hierarchy. One final differ-
enceaffectingnotourconceptionofitbutonlytheaxiomatizationisthatthe
axiomsIhaveusedfocusfirstonthenotionofalevel,anddeducetheproperties
ofsets(assubcollectionsoflevels)onlyderivatively. Thepioneerofthisstyleof
axiomatizationwasScott(1974),butmyearlierbookusedunpublishedwork
ofJohnDerricktosimplifyhissystemsignificantly,andhereIhavetakenthe
opportunitytomakestillmoresimplifications.
Lecturersonthelook-outforacoursetextmayfeelnervousaboutthis. No
executive, it is said, has ever been sacked for ordering IBM computers, even
iftheywerenotthebestbuy;byparityofreasoning,Isuppose,nolectureris
eversackedforteachingZF. SoitisworthstressingthatZU,thesystemwhich
acts as a default throughout this book, is interpretable in ZF in the obvious
manner: the theorems stated in this book are, word for word, theorems of
ZF. SoteachingfromthisbookisnotliketeachingfromQuine’sMathematical
Logic: youwillfindnoself-memberedsetshere.
Most of the exercises have very largely been taken unchanged from my
earlier book. I recommend browsing them, at least: it is a worthwhile aid to
understandingthetext, evenforthosestudentswhodonotseriouslyattempt
todothemall.
Itisapleasure,inconclusion,torecordmythanks: toPhilipMeguireand
Pierre Matet for correspondence about the weaknesses of my earlier book;
to the members of the Cambridge seminar which discussed chapters of the
new book in draft and exposed inadequacies of exposition; and especially to
Timothy Smiley, Eric James, Peter Clark and Richard Zach, all of whom
spentmoretimethanIhadanyrighttoexpectreadingthetextandgivingme
perceptive,judiciouscommentsonit.
M.D.P.
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Contents
I Sets 1
IntroductiontoPartI 3
1 Logic 6
1.1 Theaxiomaticmethod 6
1.2 Thebackgroundlogic 11
1.3 Schemes 13
1.4 Thechoiceoflogic 16
1.5 Definitedescriptions 18
Notes 20
2 Collections 21
2.1 Collectionsandfusions 21
2.2 Membership 23
2.3 Russell’sparadox 25
2.4 Isitaparadox? 26
2.5 Indefiniteextensibility 27
2.6 Collections 30
Notes 32
3 Thehierarchy 34
3.1 Twostrategies 34
3.2 Construction 36
3.3 Metaphysicaldependence 38
3.4 Levelsandhistories 40
3.5 Theaxiomschemeofseparation 42
3.6 Thetheoryoflevels 43
3.7 Sets 47
3.8 Purity 50