Thomas Forster Reasoning about Theoretical Entities Advances in Logic, 3 2003.pdf

7/29/2019 Thomas Forster Reasoning about Theoretical Entities Advances in Logic, 3 2003.pdf

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A dv an ce s in Logic V ol . 3

Reasoning AboutTheoretical EntitiesThomas Forster

2 + 2 = 4{1 ,1 }^ {1 ,1 } = {1,1,1,1}


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Reasoning AboutTheoretical Entities


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Advances in LogicSeries Editor: Dov M Gabbay FRSC FAvHDepartment of Computer ScienceKing's College LondonStrand, London WC2R [email protected]

PublishedVol. 1 Essays on Non-Classical Logic

by H . WansingVol. 2 Fork Algebras in Algebra, Logic and Com puter Science

by M. FriasVol. 3 Reasoning about Theoretical Entities

by T. Forster
mailto:[email protected]:[email protected]


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Advances in Logic - V o l. 3

Reasoning AboutTheoretical Entities

Thomas ForsterUniversity of Cam bridge, UK

V f e World Scientific New Jersey London Sineew Jersey London Singapore Hong Kong


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Published byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA o ffice: Suite 202, 1060 Main Street, River Edge, NJ 07661UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

REASONING ABOUT THEORETICAL ENTITIESAdvances in Logic V olume 3Copyright 2003 by W orld Scientific Publishing Co . Pte. Ltd.All rights reserved. This book or parts thereof, may not be reprodu ced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-567-3

Printed in Singapore by Mainland Press


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P r e f a c e

In this essay I am attempting to give a clear and comprehensive (and comprehensible!) exposition of the formal logic th at un derlies reduc tionisttreatments of various topics in post-nineteenth-century analytic philosophy. The aim is to explain in detailin a number of simple yet instructivecaseshow it might happen that talk about some range of putative entit iescould be meaningful, have truth conditions and so on, even if those entitiesshould be spuriou s. Altho ugh this ontological position has been ado ptedin relation to a wide range of putative entit ies at various times by variouspeople I develop the logical gadgetry here quite specifically in connectionwith one such move: cardinal and ordinal numbers as virtual objects andalways with the Bural i-Fort i paradox in mind.

Such a position (with respect t o num bers a t least) is one I associate w iththe work of Quine ("The subtle point is that any progression will serve asa version of number so long and only so long as we stick to one and thesame progression. A rithm etic is, in this sense, all there is to num ber: thereis no saying absolutely what the numbers are; there is only ari thmetic")though I think it is associated in the minds of many others with Dedekind.Indeed it seems to me to be wider than that, and to be an implicit partof the trad ition . So implicit, and deemed pe rha ps to be so obvious, th a tnob ody as far as I know has bothered to spell i t ou t. Th is derelictionhas had bad consequences. In my experience the gravest of these is th atthis treatment of numbers as virtual objects is confused with a much morewidely (mis)understood position, namely the view associated with Fregethat natural numbers are equivalence classes of sets under equipollence. Idem arca te this from t he p osition I explain in this book no t because I believeit to be mistaken, but merely because it invites confusion with the positionI am setting out to explain.

Th us we are com m itted to developing a logic of virtu al objects. Th epartic ular case th at I shall have perm ane ntly in mind in this essay is ordinal

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2 Preface

num bers and the Burali-Forti parad ox. Th e Burali-Forti parado x, morethan any of the others, requires for its resolution a proper treatment ofvirtual entities. I t is only by treating (ordinal) numbers as virtual entitiesproperly that we understand what the mistakes are that gave r ise to thatparadox. And mistakes I believe there have been, since I do not belong tothe school of thought that thinks that paradox is part of the web of belief.Thanks are due to the University of Genoa for inviting me to give thelectures in 1992 that were to become the start of this book and to DrGius eppe R osolini for organising my trip . I am also grateful to th e University of California at Irvine for three months sunsoaked leisure in theirspring quarter in 2001 that I was able to devote to work on this book, andto Pen Maddy, Aldo Antonelli and Barbara Atwell for making it possible.It is a pleasure also to be able to thank the Royal Society for defraying thecosts of a visit to Harvard to consult Professor Quine about these and othermatters. I would like also to thank the Victoria University of Wellington,Massey University and the University of Auckland for inviting me to talkab ou t this topic at their various sem inars. Finally Pau l Taylor earns mythanksas he has earned the thanks of thousands of logicians over theyearsfor supplying the I^Tp^X macros that print commutative diagrams.

Thomas E. Fors terDepa rtment of Pure Mathematicsand Mathematical SciencesUniversity of CambridgeWilberforce Road, CBS OWBCambridge, United Kingdom


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C o n t e n t s

Preface 11. Introdu ct ion 5

1.1 Inte rpre tatio ns 122. Definite De scrip tions 17

2.1 Form al definition of the inte rpre tatio n 192.2 Fu nctio ns from singular des cription s 222.3 Definite des criptions an d m od al realism 232.3.1 Ca rna p's t rea tm en t 253. V irtua l Ob jects 27

3.1 Congruen ce relations 273.2 Ex tend ing the language 293.2.1 Im plem entatio ns of languages with a canon ical simulation 323.2.1.1 Subv ersion 343.2.2 An illustr atio n: utilities 343.2.2.1 B ask ets of good s 363.2.3 Some stan da rd m athe m atical examples 363.3 Second-order and higher-order theories 373.3.1 Th ird- and higher-order virtu al entit ies 394. Cardina l Ari thm etic 41

4.1 T he language s of set theo ry and arith m etic 424.2 T he canonical simulation 43

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4 Contents

4.2.1 Sets of ca rdin als 454.2.2 M ultisets of cardin als an d the m ultiplicative axiom . 454.2.3 Ram sey Th eory 484.3 V irtu al illfounded sets 494.3.1 Irred und ant t rees 50

5. I tera ted Virtua l i ty in Cardina l Ari thm etic 535.1 Dou bly virtua l cardin als 535.2 M ultiply virtu al cardina ls 565.2.1 Th e pa rts th at vir tual ism cannot reach 56

5.2.1.1 The Par i s-Harr ington theorem 575.2.1.2 Exponent ia t ion 575.3 Un typed invariant arith m etic 585.4 Im plem entation -insens itivity 605.5 Ite rat ed virtu ality an d reflection 65

6. O rdina ls 676.1 T he elem entary theo ry of wellorderings 686.2 T he langua ge of ordina l arith m etic 736.3 O rdin als of we llorderings of sets of ord inals 756.3.1 W ha t does the T function do? 786.3.2 H arto gs ' theor em 826.3.2.1 Dia gon al interse ctions 836.4 Imp leme ntat ions of ordinal ar i thm etic 836.4.1 Th e Russel l -Whitehead implem entat ion and Scot t 'strick 846.4.1.1 Sc ott 's trick 856.4.2 Th e Von Ne um ann implem entat ion 85

6.4.2.1 Th e Bu rali-Forti pa rad ox for Von N eu m an nordinals 87Bibliography 89Index of Definitions 92Index 93


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C h a p t e r 1I n t r o d u c t i o n

Although I am using the Bural i -Fort i paradox as a case study, a kind of pegon which to hang observat ions about reduct ionism, this is not a history ofthe paradox, nor a picture of philosophy of mathemat i cs as it was at thet ime the paradox emerged. Somebody ought to write such a book. Untilt h a t is done, readers who want that should consult Moore [1982]. This essayis a thorough deconstruct ion of the Bural i -Fort i paradox in a mainst reampost nineteenth century logical tradition. It does not claim to be radical,but merely to make clear what the received view on this topic would be ifthe consensus of informed opinion had t roubled to formulate itself. Whywrite such a book if it has no radical insights to offer? Be cau se alth ou ghthe analysis is elementary, it is neither obvious nor particularly easy. Ifit were easy it would have been done long ago, and it were obvious therewould be no need to do it at all.

W h e n I went to university I star ted doing a degree in philosophy. Inthose days we were exposed to logical positivism: later generations broughtup on different and more modern (or post-modern) isms may scoff, butfor those interested in unders tanding how the world works it is probablyas good a point of depar tu re as any. It was then that I encountered thedoctrine that physical objects are logical con structions out of sense data. Itseemed painfully clear to me even at that early stage that if I really wishedto even understand this thesis (let alone decide whether or not it was t rue)then I would have to learn abo ut these logical constructionswhatever theymight beso I set about learning some logic.1The doctr ine that physical objects are logical con structions out of sense

d a t a is not at all atypical in modern philosophy. Claims with this flavourare made all the t ime: "meaning is use"; the sociobiology and sociobiology-*I rem emb er be ing pa r t i cu la r ly s t ruck by the sect ion on the t h e o ry of defini t ions in

S u p p e s ' I n t ro d u c t i o n to Logic , then hot from the pres s , now recen t ly and jus t ly re issuedas a classic by Dover. [1957].

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6 Introduction

inspired literature is full of warnings that attributions of intention to genesare merely a fagon de parler; the history of ethics is replete with claims thatethical judgements "just are" linguistically transformed assertions of somequite different and supposedly simpler kind, such as commands, or perhapsroa rs of app rova l or howls of disgu st. Ac cording to (ethical) em otivismmoral judgements are syntactic sugar for expressions of disgust or approval.There are cases that make it look plausible: my favourite example for yearswas the moral pronouncements made by homophobes, to the effect that gaysex is wicked and th a t gays should be punished. It certa inly seems veryplausible that the moral judgements expressed by homophobes are merelyexpressions of revulsion. So pe rha ps som ething analogo us goes for m oraljudgements in general .Plausible though this looks, I have come to the conclusion that it iswrong. Homophobes are indeed expressing disgust, but they do not therebysucceed in expressing a mo ral judgem ent despite their use of moral languageto do it! What is going on is not (i) that they are using moral languageappropriately and emotivism is true, but rather (ii) that they are usingmoral language in a nonstandard way, borrowing it to lend force to theirutterances, and emotivism is false. Any discourse can be used metaphorically, and a discourse that has power riding on it is more likely to beused metap horica lly th an o thers! So th e the superficially m oral discourseof homophobes does not give us a fact about the content of moral language(which would be interesting) but merely a fact about the uses to which itcan be put. And when you reflect that metaphor is inexhaustible, it hardlycomes as a surprise to discover that the rhetoric of morality can be usedto express hom opho bia. Th is is a fact ab ou t homo pho bes, not m orality:it is not a fact ab ou t abo ut m oral language ( th at it ha s no pro positiona lcontent) but a fact about the way that homophobes have of using it.

After all, if I use the lang uage of widgets me taph orica lly to ma ke a p ointabo ut ga dgets , this doesn ' t m ean th at widgets are just s yntactic sugar forgadgets. One must distinguish between the sentence and its uses.Even now that the legacy of the Vienna school has decayed there remains to us a feeling that we should try to think of each subject in thefollowing list as being a part of its predecessor simplified with contextualdefinition: physics, chemistry, biochemistry, psychology, economics, history.This book is aimed at people who want to make a start on finding out whatthese claims mean. My aim in writing it has not been primarily to defend

a reductive account of ordinals but to save others the bother of workingout precisely what logic-chopping such a reduction involves. I am no reduc tion ist: for m e redu ction ism is a str ate gy for flushing out ontologicalcom m itmen t. I share with the anti-reductionists a hunch tha t reductionism won't work. What I do not share is their superstition that it is possible


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Introduction 7

to understand the l imitations of reductionist s trategies without actuallyacquiring enough logic to formally execute them. This is an error: the belief that something won't work is not automatically a reason for not tryingit, for even if failure is certa in th e ma nn er of it migh t be instru ctiv e. Inany case, as Dana Scott wrote [1962] "Nothing supports belief like proof".This error arises in part from a tendency to overestimate the importance ofaquaintance with facts and underestimate the importance of skills. This isthe error of thinking that all knowledge is knowledge-that not Jcnowiedge-how. It is an error with beguiling appeal, particularly to those who, forreasons of laziness or incapacity, are averse to the effort involved in concretising their understanding of reductionism, but it is an error nonetheless.The anti-reductionists differ from me not in our beliefs about reductionism,but in the lengths they are prepared to go to test them. Unwillingness toact on this hunchtogether with the need for post hoc justifications of thatunwillingnesscall forth from the anti-reductionists a range of (post hoc!)a priori explanations for the supposed vanity of the enterprise (physicalistexplana tions canno t cap ture quaJia etc. , etc. , . . . ) , one and all spurious.The only way to understand what reductionist s trategies can (and cannot)do is to learn how to execute them . Th is is not a fact ab ou t philosophy,but a fact about the human brain and how it acquires knowledge.

One effect of the tendency of some anti-reductionists to dismiss reductionism out of hand is a failure to explain properly what it is that is beingdismissed, so that newcomers to the debate never really discover what is atissue. A recent tex tbo ok sum ma rises reduction ism as follows: "Red uction ,as it is understood in most philosophical accounts involves deriving onetheory from another, where theories are construed as sets of laws" (Bechtelet ai. [2001]) (The use of the word 'construed' flags this text as a piece ofphilosophical discourse, but the expectations it raises are not met.)This is not to say that there cannot be principled reasons for believingthat reductionism is a vain enterprise. There may be. I suspect that theimpulse to believe it is doomed stems from a cause whose best illustrationis the old saw that one cannot derive an 'ought' from an ' is ' . This is indeedstarkly obvious, and to those of us brought up in twentieth-century logicthe obviousness seems to arise from the interpolation lemma. This statesthat if we have derived a formula Q from a formula P, then there is aformula R. derivable from P. and from which Q in turn can be derived,which contains only vocabulary common to P and Q. The simplest caseof the interpolation lemma concerns the case where P > Q is a tautologybu t P and Q have no nonlogical vocabulary in common. In this case eitherQ is a tautology or P is the ne gation of one. And this case is the oneHume had in mind: after all, the vocabularies of physics and of moralisingare disjoint, and so only in trivial cases can one deduce an 'ought' form


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8 Introduction

an ' is ' . Th e interp olatio n lemm a is one of the most intuitively obviousresults of modern logic, and the intuition which makes it obvious is at workeven in people who know no logic, and in them it results in a convictionthat reductionist strategies must fail. And yet, one wants to say, it is trueboth that everything has to emerge from 'is 's and that we can't just writem orality off. On e way round th is is to atte m pt to interp ret m oral langu agein physicalis t language. One cannot derive an 'ought' from an ' is ' but onemight be able to interpret an 'ought ' as an ' is ' . 2

The conditions controlling use of a contextually defined term can takethe form of syntactic constraints . These syntactic constraints can in turntak e the form of typ e-the oretic distinctions of the kind th at were first explicitly developed by Russell [1905]. Since that time a type-theoretic hierarchyof some kind has traditionally been invoked as a way of dealing with theparadoxes that so plagued the foundations of mathematics at that t ime.It does not nowadays take the genius of a Russell to see that this was agood idea. Indeed in contrast nowadays (and especially since Kripke [1976])we are so biase about this insight that it now almost seems to be a goodidea whose time has gone: it has become fashionable among philosophersto decry type-theoretic cr i t iques of the paradoxes. Although an approachthat eschews type-theory looks quite plausible for the semantic paradoxes(which are those that philosophers are more likely to be concerned with)it does not work nearly so well with the logical paradoxes (and the Burali-Forti paradox in par t icular) , which have much deeper roots in hierarchytheo ry. It is no t har d to see how this fact cam e to be overlooked , forthe Burali-Forti paradox is by far the most obscure of the paradoxes: evenBu rali-Forti, when w riting th e article (Bu rali-Forti [1897]) in virtu e of whichthe paradox bears his name, did not think that he had discovered a proofof a contradiction, but a reductio ad absurdum proof that ordinals are nottotally ordered. Howeverand this important point will become clearer asthis essay proceedsthe type-theoretical hierachy that we need to exploitin order to avoid the Burali-Forti paradox has i ts roots in the syntacticdisciplines entailed by contextual definitions, and has nothing to do withthe type distinctions of Russell [1905].

The logical positivists used the expression 'category mistake' to capturesome of these odd ities. My favourite example is C arn ap 's illustration : "Thisstone is thinking about Vienna" (which reveals where the example comesfrom!). "X is thinking about Y" is (or can be analysed as) an abbreviation2 Para l le ls can be found in th e l i te rat ure in ha rdw are ver i f ication . Co rrectn ess asser

t ions seem to belong to a d i f feren t vocabulary f rom that used to descr ibe ci rcu i t ry , andthe in ter pola t ion lem m a makes it sel f fel t again , the effect th at co rrectne ss proofs canbe of no effect . A nd a ga in, the re spo nse is to look for ways of int er pre tin g the secondvoc abu lary in the f i rst . See M elha m cha pter 4 , and E veking , [1991]


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Introduction 9

for a long conjunction of assertions about X. The point is that the linguisticrules will allow this abbreviation to be introduced only if certain things aret rue , and t he vast majo rity of tho se thin gs are false of ston es. I free-associate from this quotation to memories of discussions of the truth-valuestatus (True? False? Meaningless?) of such monsters. I shall not be joiningsuch a debate here: my purpose is to show how the Burali-Forti paradoxcan be seen as arising from a failure to respect the conditions attached toa contextual definition in a similar way, and partakes of the absurditynot just of the other paradoxesbut also of the assertion that a stone isthink ing ab ou t Vienna. In some ways this is a mo re instructiv e pas time :although the conventions surrounding the use of ordinal notations can bemade explicit in a way that the conventions breached by talk of a stonecontemplating Vienna cannot (for the moment!) they have not so far beenmade thus explicit .

There is another cause of people overlooking the deeper roots thatBu rali-Forti has in type-theo ry. For m any years I for one believed th attype-theory was primarily a way of avoiding the paradoxes. It was not untilI started teaching computer science and was forced to think about a distinction that for want of a better word one might call an essence-implementationdistinction that I came to realise that the purpose of respecting type disciplines is to avoid making fools of ourselves by overlooking it. If we respecttype disciplines in our use of language, then we can avoid asking silly questions like: "Is there an x identical with the ordered pair {x,x)l" Althoughthere is a risk of self-centredness in assuming that others will have madethe same mistakes one has made oneself, one can be very useful to themif one happens to be right! It now seems to me that type-theory plays amuch more central role in the dissolution of the Burali-Forti paradox thanit does in any of th e othe rs. Indee d I would claim th a t it is only a typ e-theoret ic cri t ique of the Bu ral i-Fort i paradox th at enables us to u nde rstandit at all . By "type-theoretic" I do not mean 'type-theory' as in what themodern radicals ridicule as the "Tarski hierarchy" but ' type-theory' as inthe syntactic disciplines we observe in order to avoid impaling ourselves onquestions like "x = (x, x)T", namely "type-theory" as understood by peoplein theoretical computer science.

The conclusion of this l ine of thought is that the modern heresy thattype-theoretic approaches to the paradoxes are wrong (and incoherent) isno t likely to be fruitful. If th er e we re any poss ibility of freeing ourse lvesfrom type-theory altogether there would be some point in the thought-experiment of approaching the paradoxes armed instead with some otherweapon. But since we are always going to have to respect type distinctionsof some kindif for no other reason than that type-distinctions are forcedon us by contextual definitions (and it is those type dist inct ions that are


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10 Introduction

at the root of the Burali-Forti paradox)we may as well try to put themto good use.To understand the role of contextual definitions in the manufactureof theoretical entities it is an idea to start with a case we understand.First-order cardinal ar i thmetic is probably the most complicated instanceof this phenomenon for which there exists a community of scholars whounderstandin outl ineits s tatus as a theory of contextually defined enti t ies . Higher-order ordinal ar i thm etic is probably th e most c omplicatedinstance of this phenomenon for which it is possible to give an exhaustiveaccount at this stage. Instances of wider philosophical interest are unfortunately too complicated to treat properly for the time being. The reason forplacing the Burali-Forti paradox in a general philosophical context in this(what might be felt to be tendentious) way is that although the Burali-Fortiparadox is of considerable independent interest it can also be thought of asa toy version of the puzzle of contextually defined entities which permeatesanalytical philosophy.

However the re is one respec t in which it is of greate r inte rest. Th ereare several suites of entities admitting reductive analyses arsing from congruence relations which are generally well un der stoo d. One think s of thevarious standard presentations of integers over naturals, rationals over integers, reals over rationals, vectors as equivalence classes of directed linesegments (see Lederman-Vajda op. cit. for exa m ple ). In all thes e casesthe equivalence classesindeed even the quotients themselves, the sets ofequivalence classesare sets according to even the most penny-pinchingcomprehension axioms, so one never has to concern oneself with the dif-ference between simulations and implementations which will be importantbelow (and which will inevitably be important in other context such as philosophy of mind where one might want a reductive story that really does saythat certain things do not exist) . The other difference is that any reductionth a t gives us cardin als, will also give us sets of card inals. T he result ofthis is that there will be cardinals of sets of cardinals: the reduction can beiterated. As we will see below, this has important ramifications.

SummaryWe star t with a quick survey of interpretations, rapidly restr ict our attention to interpretations that arise from congruence relations, (but not beforewe have had a brief look at the manner in which definite descriptions canbe added to a theory obtain a new theory interpretable in the old) and seehow these can give rise to first- (and higher-) order theories of the originalenti t ies with new vir tual enti t ies . A dist inction is m ade between interpretat ions that respect equali ty and those that do not. Two examples (cardinal


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Introduction 11

and ordinal arithmetic) are pursued in some detail . Some time is spent onsketching which parts of what is commonly regarded as cardinal and ordinal arithmetic can be interpreted back into the set theory from which theyarise in a manner that does not depend on the ways in which cardinals andordinals are imp lem ented. T his leads to some discussion of set-existenceissues which, although peripheral to the rhetorical purpose of the essay, arenevertheless of some interest to students of philosophy of mathematics.

Fin ally a word or two is in ord er on w ha t is not being done. Although theendeavour is to shed some light on how assertions apparently about thingsthat do not exist can have truth-conditions and a sensible semantics, thistre atm en t is not intend ed to explain how state m en ts ab ou t fictional figurescan have genuine truth-conditions (or not, as instance L.C.Knights famousquestion "How m any children ha d Lady Ma cb eth ?" ) nor is i t intend ed tocover truths about nonexistent objects of the kind considered by Condo-ravadi et al.Historical remarksThere is no doubt that the Burali-Forti paradox is the most obscure of theclassical paradoxes. Indeed, at i ts first appearance it was not even recognised as a paradox, but believed to be a reductio ad absurdum proof thatordina l inequality is not trichotom ous . (This is the neg ation of theo rem6.4.) Ru ssell 's view in 1901-3 wh en he was writing his Principles of Math-ematics [1903] (see ch XXXVIII paragraph 301) was that i t proved thatthe o rder relation on ordinals is not wellfounded. (Th is is theo rem 6.4.)By the t ime he was wri t ing Principia Mathematica with Whi tehead, h isview had changed, and he had come to the modern view. (See volume IIIof Principia Mathematica pp 7 4-5). Eve n as late as 1940 it wa s possiblefor as distinguished a scholar as Quine to publish an axiomatisation of settheo ry in which an imp lem entation of the para do x could be proved. (Seep . 85.) T he fact tha t the stan da rd proofs of all the other pa rado xes ha dbeen successfully blocked underlines how obscure the Burali-Forti paradoxis . Russel l (Introduct ion to Mathematical Phi losophy) doesn ' t mention i tat all.

NotationWe use stan da rd set- theoret ic notat ion he re. 0 is the em pty set , [jx is{y : (3z)(y z A z x}, f)x is {y : (\/z)(z x > y z}, f(x) is the valuethe function / gives to the argument x, to contrast i t with f"x which isth e set of values the function / gives to th e set of arg um en ts x. We willsometimes omit the brackets from functional application where a A-calculus


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12 Interpretations

trad it ion (which eschews them ) seems more stylish. Th us ' f s t ( x , y) ' rathertha n ' f s t( (a; , y)'). Xx.[...} is the function th at , on being given x, re turns thestuff enclosed by the square brackets. A superscripted arrow on a variablemeans a list of variables, thus: x. ' ' signals the end of a proof. FollowingRussell, we use upside-dow n iotas for singular desc riptions . Exp ressionswrit ten in b o l d are being defined; expression in italic and being emphasised.Quine quotes ("corners") are a device for obtaining variable names forexpressions with variables in the m . Th ey should be thou gh t of as a diacriticwhich creates a context within which names of expressions can be combinedwith connectives to give names of the expressions thus combined.We will use upper case Ro ma n letters for var iables ( 'X ' , ' X " , ' V . . . )

ranging over sets. Lower case variables (lx\ lyn, 'y ' . . . ) will rang e overwh atever it is th a t those sets are sets of. We will have these tw o stylescommon in set theoryso that we can write 'x X' as usua l. C apitalisedfraktur font variables will range over s t r u c t u r e s . If X i s a mathematicalstructure consisting of a set X ( the d o m a i n ) and a relation R on X we w illsay tha t X = DM(X) and R = DG(X), and the letter denoting the domainof a s tructure will be the uppercase Roman letter corresponding to theuppercase fraktur let ter denoting the structure. A r e d u c t i o n of a s truc tureis a s tructure with the same domain but some operations or predicatesremoved. An e x p a n s i o n of a s tructu re is a s tru ctu re with th e same do ma inand operations with some extra ones added. See (e.g.) Hodges [1993]

1 . 1 I n t e r p r e t a t i o n sAn interpre tation I of a theory T in a theory S is a recursive map fromC T to C s which sends elements of T to elements of S. By 'recursive' Imean that i t should be denned by recursion on the grammar of C T ra therthan that , considered as a map from Godel numbers to Godel numbers, i tshould have a decidable graph. Announcing this restr iction shouldn ' t s tar tany fights: there may be circumstances in which nonrecursive maps need tobe taken into account, but they destroy information too comprehensively tobe any help in an analysis intended to allow understanding to flow throughthe interpre tation from th e interpreting language to the interpreted. In factthe nature of the interpretations we consider will be restricted still further.

Clearly I must send atomic n-place relation (resp. function) symbols ofC T to (possibly derived) n-place relation (resp. function) symbols of C s-Typically it will be the case that C T is a superset of C s- For example C smight be the language of set theory, and C T the language of set theoryplus ar i thm etic. Again C s might be the language of physics, and C T thelanguage of physics plus some mentalistic language.


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Introduction 13

X might satisfy other sensible conditions, for example it might send =to = . The condi t ion that X be recursive suggests (though it does not entail)t h a t X should commute with quantifiers and connectives. (If X is recursivewe know th at X of \? A $ must be (X(\&))??(X($)) for some binary connective ??.) One example of an interp retation th at com mu tes in this way isthe interpretation of meta-ar i thmetic into ar i thmetic that is needed for theproof of the incom pleteness theo rem . A family of exam ples of inte rpre tat ions that do not so commute are the negative interpretations of classicallogic into intuitionistic logic. Others include the Godel interpretation of amodal proposit ional language in a nonmodal one and the interpretation ofa classical dyadic language into an intuitionistic monadic one.

Whether or not an interpretation sends atomics to atomics or commuteswith logical op era tors is not going to ma tte r a gre at deal here. Alth oug hwe will distinguish two kinds of interpretations we will distinguish them ondifferent grounds: those I shall call implementations and those I shall callsimulations.An interpretation of L\ into 2 that sends equality to equality is animp l emen t a t i o n . All other interpretations are s imu l a t i on s .When we have a simulation of a language C\ into another language 2we shall say that from the point of view of 2 the entities of 1 are v i r tua l .In the situations that concern us here we will be interested in simulations where the interpretation of equality is a equivalence relation which isna tura l in tha t contex t. For exam ple we will s imulate cardinal ar i thm eticin set theory by interpreting equality between cardinals as equinumerosity(aka equipollence) between sets. Such simulations we will call canonical.The situations in which one looks for interpretations fall naturally intoone of two classes, depending on whether on not the entities one wishes toimp leme nt a re to be "m agicked away" or no t. A familiar ex am ple of thisis the interpre tation of elem entary-ar ithm etic-with-pair ing-a nd-unp air inginto elementary ar i thmetic. One can implement (x,y) as 2X x (2y 4- 1) or(x+2+1) + x or in any of a host of other ways. In set theory it is customary to interpret settheory-with-pair ing-and-unpair ing inside set theory bythe Wiener-Kuratowski device of taking (x,y) to be {{x}, {x,y}}, thoughit is reasona bly well known th at the re are other w ays of doing it. Russellfamously showed how to interpret languages with singular descriptions intolanguages without singular descriptions. In neither of these cases is thereany suggestion that ordered pairs are not real, or that the things singularlyidentified are not real. In these cases the interpretation involved is an implementation. This is in contrast to cases (like those that will occupy mostof this book) where the language being interpreted speaks of entities thatare not present in the language into which the interpretation is directed,and the interpretation is a simulation. Those are the cases where there


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14 Interpretations

seems to be a genuine possibility of ontological reductio n. "Th ere are nonum bers, there is only ar i thmetic." ( to misquote Qu ine) .In circumstances like this it is natural when considering expressionsin the language of (for example) set-theory-with-pairing-and-unpairing toascribe a privileged pos ition t o those expressions (j) such that the truth-valueof the implementation of cj> into set the ory does not d epe nd in any way on theimplementation (of pairing and unpairing) chosen. It 's fairly plausible thatthese expressions can be given a purely synta ctica l cha racte risation . Th eexpressions that fail the syntactic test (like (3a;)(a; = {x,0})) also have theflavour of puns, but this time for a subtly different reason. The point is notthat the canonical simulation is not defined on them (true though that is,since there is no canonical simulationindeed no simulation at all) but thattheir truth-value is imp leme ntation-depend ent. Pun ning of this kind is animportant and idiomatic part of some programming languages (specificallyand impor tan t ly the languages C an d C++) and is the most impor tan tsingle cause of the distaste with which these languages are regarded by thecognoscenti. It also plays a central role in the proofs of the major resultsof recursion theory, most obviously the proof of the unsolvability of thehalting problem, which relies on a number being both (i) treated as dataand (ii) executed as the index of a machine. Much of what there is to sayabout the Godel sentence does not depend on a choice of Godel numbering.This is less true for Rosser sentences, and is one of the reasons why Rossersentences are less well understood.

This semantical character isation can be made also in caseslike thatwhere L\ is the language of set theory and cardinalswhere there aregenuine inte rpre tatio ns in additio n to a canonical simu lation. (Th ere isobviously no cano nical simu lation in th e case of orde red pairs!) W ill itturn out that the subset of the language of set theory and cardinals onwhich the canonical simulation is defined is the same as the subset which isimplem entation-independ ent? Irr i tat ingly not, though th e f irst is at leasta subset of the second. We shall see exam ples of expressions th a t areimplementation-independent but still not in the domain of the canonicalsimulation. An important example is the combinatorial asser t ion about INth at l ies at the heart of the Paris-Harr ington theorem . Othe r exam plespave the path to the Burali-Forti paradox.Outside the strictly mathematical context which will be the main concern here we can see other areas where this s imulation/implementation

term inolog y will be useful. P a rt of th e m od ern function alist rescension ofphilosophy of mind is a view that mentalistic language is always an optionwhich is available for describing in neat shorthand the gross behaviour ofsystems that exhibit (sufficiently complicated) feedback. For example Dennett ( in [1979]) writes of "adopting the intensional stance". In the slang of


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Introduction 15

computer scientists, mentalistic language is syntactic sugar for physicalisttalk. In the terminology of the preceding paragraph, we are working in an(as i t might be) physicalist language 2 and trying to interpret in i t a men-tal ist ic+physical ist ic language C\. W e simulate mental ent i t ies. What thevarious flavours of identity theory currently available in the philosophicalli terature haggle over is the nature of the physical implementations of themental ent i t ies that the intensional language purports to describe.

Th ere is a large class of inter pre tation s which have philosophical interest .As we have noted, there are many claims of the kind "widgets are justgadgets", and in the terminology that prevails in this book claims like thatwould come out as "The theory of widgets and gadgets can be interpretedin the theory of gadgets". However not every claim of this kind is felt tohave ontological repercussions in the way that most of us would feel thata reduction of sets-with-cardinals to sets shows that there aren't really anynumbers at all . Certainly in the simplest case of allsingular descriptions,with which we begin the next chapterthere is no suggestion that the Kingof France fails to be real merely because the expressions that denote himand him alone can be magicked out of the language: the President of Prancecan be magicked out of the language too, but he is real enough.There are many vir tual objects that one might be interested in studyingand for which a robust and taut logical analysis is both needed and available. One thin ks of a virtua l theory of waves interp reted in a physics wh erethe only real things are addresses in space-time and elementary particles.3

3 T hi s would be a c la ss ical theory no t a qu an tum theory! In qu an tum theory w avesare not vi r tua l wi th respect to part ic les : they are equal ly rea l .


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C h a p t e r 2

Def i n i t e Desc r i p t i ons

Although in most of this book we will be preoccupied with interpretationsthat arise from congruence relations, there is a very simple and very important example of an interpretation which we will need to master first , andthis is Russell 's treatment of singular descriptions. And we need to do thisnot merely because it is simple, or historically prior to other such analysesthough it isbut because we need to understand singular descriptions inorder to give a smooth treatment of functions defined on virtual entit ies.Readers who are completely confident about Russell 's theory of descriptionscan probably safely skip the rest of this chapter.

In Russell 's treatment of definite descriptions we replace expressionsmatching the template(3x)(ip(x) A (Vy)(^(y) - y = x) A {x))

with expressions matchingMKxWix)))

(The vagueness in this description is deliberate: see below). As happens sooften in Philosophical Logic, the analysis he gave can be used in two ways.On the one hand it can be used to develop formalised languages so thatthey exhibit features which behave in various helpful ways like features ofnatural languages (i.e., in this case it explains how to introduce definitedescriptions into formalised languages). On the other hand it can be usedto make a claim about what the logical status of those natural languagefeatures really is. Th is distinction is som etimes called the prescriptive-descriptive distinction. Philosop hers often take Russell 's analysis in thesecond sense, thereby regarding it as a (descriptive) thesis about ordinarylanguag e to which reasonable people might take excep tion. (Strawso n [1950]is an examp le.) Ou r prim ary concern here is with formalised language s

17


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18 Definite Descriptions

(since part of the aim of the project is to illuminate the Burali-Forti paradox) so this second point of view will be touched only briefly.In addition to the two uses to which Russell 's analysis can be put, thereare also two views one can have abo ut w hat it actually says. Th e sam etwo possibilities exist in relation to the indefinite descriptions of Hilbert,and there the possibilities of confusion are much more serious. It thereforeseems a good idea to deal with the matter when it f irst comes up, when thecontext is simple.On the one hand we can think of v-terms as being introduced by contextual definition, so that (p((\x)(ip(x)) is short for (3x)((y) > y = x)). According to this version of events, the introduction

of v term s is an entirely orth ogra phic al m ove, and is noth ing to do withlogic at all. If one wants to think of Russell's account of v-terms as a thesisabout the logical structure of ordinary language, this compels one to saythat the (top level) logical structure of "the King of France is bald" is anexistential quantification. According to this view, r-terms term s are syntac tic sugar and do not really denote. However, we are not obliged to adhereto this purist view: in any situation in which an r-term can legitimatelybe introduced there is a canonical implementation for it. This is in radicalcontrast to other contextually defined terms, as we shall see later.On the other hand we can think of v-terms as constants in a new language, belonging to a new theory which is a conservative extension of theold ("no new theorems in the old vocabulary"). The completeness theorem for first-order logic tells us that if we have a theory T expressed ina first-order language, and T proves an expression of the form (3x)ip(x)the extension T" of T obtained by adding a new constant to the language'a' , and adding an axiom ip(a) to T is conservative. Therefore the coher-ence of this view of singular descriptions for a language C depends on the

completeness theorem for C Applying this version of Russell's analysisprescriptively to the examination of ordinary language we find that it tellsus tha t "the King of Franc e is bald" really has the structure of a predicatebeing attr ibuted to a subject , not that of a existential quantification.The precise status of claims like these is obscure: no-one can seriouslypretend that these abbreviations are explicitly set up by a properly constituted parliament of language users. One rapidly finds oneself facing thesame problems as social contract theorists do in ethics.Although Strawson's cr i t ique is based on the assumption that Russell 's

reduction of singular descriptions is to be taken as an account of singular descriptions in ordinary language and therefore is not directly relevanthere, this point of view has a formal correlate. The idea that "The King ofFrance is bald" lacks truth -valu e when th ere is no unique king of France corresponds to a determination not to allow expressions containing (vx)((j>(x))


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Formal Definition of the Interpretation 19

to be well-formed when there is not precisely one thing that is . Normallyin m athe m atical logic one sets up a language and then on th e und erstan ding that this language is fixedone adumbrates theories in i t . In principlethere is nothing to stop one having an arrangement in which the languageis regarded as a dynamic entity, so that the language belonging to a theoryis determined in part by that theory. In this case one would allow that ifT h 3\x(p(x) then a constant ' ra.0(x) ' can be added to the language westarted with. This may or may not have the merits of psychological plausibili ty and exciting technical complexity but i t is not a path I intend togo down he re. T he o bvious question of w ha t one is to ma ke of formulaecontaining embedded definite descriptions whose introduction has not beenauthorised by a theorem in the style 3\x{x) is largely sidestepped here bytheorem 2 .1 .

In general (3\x)(3\y)(f>(x,y) is not the same as (3\y)(3\x) C defined as follows.Th e recursive clauses are th a t I com m utes with quantifiers and con nectives.Atomic and negatomic formulae are sent to themselves unless they containsingular descriptions. Let us consider this case in more detail .

If 'x' is free in the n for atom ic or ne gato m ic ^ w ith ' y' free, 2" ofty{vx.(x) A *[x/y] A (Vy )(0(j /) -> y = x))If \I> is an equation this last expression becomes

(y) A (Vz)((z) - z = y)It is important that the rewriting rules we are implicit ly defining by

this declaration should be confluent! The expression


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It doesn't make any difference which order these two expansions aredone in.If we treat l4>((?x)A(x),(vy)B(y)y as an assertion about ( w = x) A w = x)AB(y)A(Vz)(B(z) -> z = y)A4>{x,y))... rear range the matr ix(3y)(3x)(B{y)A(Vz)(B(z) - z = y)AA(x)A(Vw)(A(w) - w = x)A0(x , t / ) ). . . import the '3a; '(3y)( JB(2/)A(V2)( JB(z) - z = y)A(3x)(A(x)A(Vw)(A(w) -> u; = x)A4>(x,y)))abbreviate the third conjunct

(3y)(B(y) A (Vz)(B(z) -> z = y) A (x)/y] is logically equ ivalent to

(3x){(f>(x) A V[x/y} A {Vy){ct>{y) -> y = x))Proof:This is proved by induction on *f>. The base case (vp atomic ornegatomic) follows immediately from definition 2.1. The inductive stepsfor the quantifiers and connectives are as follows.

$ is tf i V # 2


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Formal Definition of the Interpretation 21

Suppose 'x' is free in

i or \I>2- By induction hypothesis '9i[^x.(j>(x)/y\ is equivalent to(3x)(4>(x)A^i [x/y]A(\/y)((j)(y) > y = x)) and \I>2[rx.(/>(x)/y] is equ iva len t to(3x)(0(x)A# 2[x/y]A (Vy )( y = x)) . Therefore (* i V* 2)[rx.((:r)/y]is equivalent to

(3x1)( y = x))as desired

# is * i A * 2Suppose 'x' is free in and no variable free in (f> is bound by any quantifier in $i or \I>2. By induction hypothesis iSi[^x.(j>(x)/y] is equivalent to(3x)(0(x)A\I/i[x/y]A(Vy)((y) > y = x)) and \E r2[^- y = x)). Therefore (* x A*2)[rx.


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can push the 'Vz' further inside to get (3x)(4>(x) A(Vz)^[x/y] A(Vj/)(0(j/)y = z))The case with the existential quantifier is similar but easier.

For what it 's worth this proof is constructive. Notice that this inductionwill no t wo rk for -> an d >. Formu lae bu ilt u p from t he se co nn ectiv es arenot covered by this result. However every formula is classically equivalentto one that is . We need the clause that no variable free in is bound byany quantifier in ty for the following reason. If we tak e (x) to be "x isthe mother of z" and * to be l{iz){y is related to z) ' then the result ofapplying the above rule to '$>[rx. y = y')) then clearly we are justified in introducing thenotation for the function connoted by the letter

ltp'. Notice that nothingin theorem 2.1 depends on containing no free variables. Thus 2.1 applies

to function letters as well as to singular descriptions, since a ( m o l e c u l a r )func t i o n s y m bo l i s s i m pl y a de f i n i t e de s c r i p t i o n w i th f r e e v a r i -a b l e s .Henceforth we will assum e w ithou t further com m ent th at function letters can be introduced in this way. We will make use of it often in what isto follow, but always in circumstances where the value of the "function" isnot unique but is unique-up-to-equivalence under the relevant equivalencerelat ion.In the long run i t is important to understand the correct t reatment ofsingular descriptions with parameters because that is one way of avoidingfunctional notation. In the short run, it enables us to make sense of an oldpuzzle of Quine's to which we now digress.


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Definite Descriptions and Modal Realism 23

2.3 Def ini te desc r ipt ions an d m od al real i smQu ine's puzzle is as follows. Consider the two term s '9 ' and ' th e nu m berof pla ne ts ' . One might suppose they den ote the sam e thing , but if the ydo, then anything true of (the denotation of) one is true of (the denotationof) the othe r. Un fortunate ly the first is necessarily grea ter tha n 7 andthe second isn't. So either they aren't the same thing, or substitutivity ofidenticals fails.To resolve this, we will have to ascertain the logical structure of "Thenumber of planets is necessarily greater than 7". What is its main connective? Is it a box? Or is it a predication. If it is a predication we can unravelas follows:

A: ((ra)(number-of-planets( :c)) > 7)appears to ma tch the pat tern ^{(^x)((f>(x))), (using the template of theorem2.1) by matching 'number-of-planets(x)' to '(x)' and 'Q(x > 7)' to l * ( x ) ' .We know what to do with this. It is just:

U(x)(3x) /\ { (Vy)(4>(y) - j / = z)l * ( s )

so A must beA ':

( number-of-planets(a;)(Vy)(number-of-planets(y) > y = x)0{x > 7)which is what we obtain by substi tuting lnumber-of-planets(a:) ' forl4>(xy and 7)' for l * ( x ) ' .The question now is, is A' actually true? The answer is: obviously it is.The witness to '(3x)' is clearly 9, for 9 satisfies all th e co nju nc ts. So A istrue as well!This is the answer modal sceptics want, since it appears to show thatif one uses modal apparatus one is eventually compelled to accept as truemany assertions that those who believe in the integrity of modal apparatuswould insist are false.The other possibility is that "Necessarily the number of planets isgrea ter th an 7" is of th e form $ (where is


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of the formD\I>

is really of the form(VW)(W (= *)

where the fragment '(VW)' is a quantifier over entities called 'possibleworlds ' .According to this view,A: D((rx)(number-of-planets(x)) > 7)

is really (unravel the box)A ": (V W )(W (= ((vx)(num ber-of-planets(x)) > 7))

Next we unravel the s ingular descript ion by appealing to theorem 2.1.Using the notation of that theorem we find 7)). Now we have to check that no variable free in"nu m ber-of-p lanets(x)" is bo un d by any quantifier in "(VW)(W (= y > 7)" .Now believers in possible worlds regard all atomic formulas as having anextra place which is occupied by a variable over possible worlds: primitiveassertions cannot be "x is F" bu t x is F at W ". Therefore "number-of-planets(x)" contains a hidden variable over possible worlds, (so that it issomething l ike "W |= number-of-plane ts(x)") and so the variable-capturecondition of the theorem is not satisfied. If this is so, then the correct wayto in te rpre t A in the language without s ingular terms is

{ number-of-planets(x)(Vy)(number-of-planets(2/) -> y = x ) )x > 7which states that in every possible world there are more than sevenplanets. Modal realists are united in agreeing that this is false.

It seems fairly clear to us nowadays thatsince it gives us an analysisof the troublesome proposition that comes out false as desiredthis is insome sense the "correct" insight to apply to Quine's puzzle. Tha t is tosay: it is th e correc t insight to a pp ly to the p uzzle if we are going totake the puzzle seriouslythat is to say, not repudiate the modal languagealtogether. Qu ine 's repudiat ion of mo dal ta lk is very germ ane to the widerissues with which we are concerned h ere, for his analysis of mo dal languag e(in [1966b]) famously allows modal operators attached to closed formulae assyntactic sugar for expressions which name those formulae and assert that


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Definite Descriptions and Moda l Realism 25

they are logically valid. Expressions consisting of modal operators attachedto the front of other formulae are not generated in this way and are notjustified by this analysis. Whether or not one is attracted by this analysisis not the point here: I am bringing it up because i t is another example ofan analysis of the kind I wish to provide later for ordinal arithmetic.Quine's puzzle at least provides us with an example to i l lustrate thefact that the rewriting rules that arise from interpretations need not beconfluent.

2 . 3 . 1 Carnap's treatmentHowever, there is another approach to this , due to Carnap, that (as wewould now put i t) swallows Quine's bait . It takes "the number of planetsis necessarily greater than seven" to be of the logical form of a predicateattached to a singular term, and yet avoids the unpleasant conclusion.The idea is to have a much stronger notion of equality, so that fewerthings are equal to each other, and it becomes correspondingly harder tofind a witness to th e existe ntial quan tifier. If we do this sufficiently tho roughly, then the expression will come out false as desired.Carnap's way of doing this (in [CI]) is to fragment all the old entities(9 , which is also the number of the planets, the sum of 5 and 4 etc) intolots of new ones, so that practically nothing is identical to anything else.In part icular there is now no ent i ty that is both the number of planets andis necessarily grea ter tha n 7. Th ere is a num ber 9, an d there is ano therintensionally identified e ntity which is the nu m ber of the p lane ts. Altho ughthere is a new equivalence relation ~ that holds between these new entit ies,it is strictly weaker than equality, so there is, as we wanted, no witness toA ' . Wh a t is t rue is that

{ number-of-planets (x)(Vj/)(number-of-planets(y) > y ~ x))a(x > 7)but since this has '~ ' instead of '=' i t is not a proper translation of A.Although many (including me) feel that Carnap's approach to this puzzle is not helpful, the idea it contains that one can clarify matters by payingclose attention to the notion of equality involved in a problematic singulardescription is a useful one to which we will return in later sections.Quine says something like this in his appendix to Carnap [1947].


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C h a p t e r 3V i r t u a l O b j e c t s

I warned the reader earlier that I am going to consider only virtual objectsthat arise from congruence relations. Althoughas indicatedthis is certainly not the only kind of virtual entity one might wish to think about i tis at least a sensible place to st ar t. R ead ers may feel th a t th is simplification is an oversimplification, and may be reminded of the joke about thephysicist who was consulted by a punter for advice on how physics couldhelp him wo rk ou t which horse to back. T he phy sicist 's research led initially to a basic theory which worked "only for a spherical horse travellingin a vacuu m". Bu t one has to star t somewhere! Restr ict ing at tent ion tointerpretations arising from congruence relations makes a treatment of thelogical issues involved relatively simple. For another, at least some of thereal-life reductionist positions belonging to the mainstream rely on equivalence relations: although the formulation "chemistry is the study of thatpart of physics for which the relations between atoms of having-the-same-number-of-protons is a congruence relation" might not be mainstream, itscon tent is close to it. Finally, ordina ls arise from an equivalence r elationon wellorders, and our m ain concern here is with th at case. Accordinglyrestricting our attention to virtual entit ies that arise from equivalence relations is not unreasonable. The time has now come to develop what I earlier(p. 13) called the c a n o n i c a l s i m u l a t i o n .

3 . 1 C o n g r u e n c e r e l a t i o n s

Fix a language C until further notice. It has pred icate letters, functionletters and constants. We may introduce extra function symbols by meansof the manoeuvres in section 2.2. It has countably many variables each ofwhich is a lower case Roman letter with a subscript from IN, which will beomitted if it is zero.

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28 Virtual Objects

We will need to define congruence relations. In what follows ~ will ofcourse be an equivalence relation. In most interesting cases the congruencerelation will be definable, i.e., captured by a formula of C.D ef in i t io n 3 . 1 (According to a theory T) an equivalence relation ~ is aco ng ruence re la t io n f o r

a predicate P iff T h (( f\ Xi ~ j/) A P{x)) -> P(y)i< nwhere n is the number of free variables in P; an n-a ry function / iff T (- (f\ x t ~ y t) -> / ( f ) ~ f(y).

i< nP and / are not assumed to be atomic, and in general they won't

be. T he cong ruenc e relation w ill give rise to equivalence classes. If wewrite our congruence relation with a t i lde: '~ ' then the equivalence classof x is customarily denoted by '[a;]^,'. In contexts where it is clear whichcongruence relation we have in mind the t i lde will be omitted.This definition has to be given separately for predicates and for functions. G ran ted , one can eliminate function letters in favour of predica teletters and singular descriptions, but the notion of congruence for functions is weaker th a t th an for pred icates . Consider congruence of integersmod 17. This is a congruence relation for the binary functions x and +,though not for the ternary relations 'x + y = z' and 'x x y = z\ However,definition 3.1 is horn. This suggests an operation: input P and ~ ; ou tpu tP l { 5 ^ P : ~ i s a congruence relation for S}. This is of course the relation( 3y )( A i j ~ yi)AP(y)). This construction is useful ifas with congruence

i< nm od 17we are working in a context w here we have an equivalence re lationand t he re is a relation of im po rtan ce to us for which the equivalence relationis not a congruence relation, and we would like i t to be. We plump out therelation {(x, y, z) : x + y = z} to the relation {(x, y, z) : 17\(x + y z)}, andcongruence mod 17 really is a congruence relation for this last predicate.This plumped-out relation is functional in the sense that for all x and ythere is a z such t h a t 17|(a: + y z) and al though this z is not unique allsuch z are congruent mod 17, as follows(3z)(x + y = z (mod 17) A (\/z')(x + y = z' (mod 17)->2 ~ 1 7 Z

1))

which will in due course justify the use of a singular term (namely x + y(mod 17)) along the lines of (vz)(x + y = z (mod 17)) in the languagewith variables ranging over integers mod 17. This is of course the way torepresent functions as definite descriptions with free variables. Notice the


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Extending the Language 29

occurrence of '~i7' instead of '=' in the formula above: in other respectsthe displayed formula matches the template of definition 2.1 exactly.

The set of equivalence relations that are congruence relations for P is achain-complete poset. If ~ is a congruence relation for P, so is any equivalence relation th at is stricte r tha n ~ . Th is is not th e case for congruen cerelations for functions! The assertion that ~ is a congruence relation for P,and that it is a congruence relation for / are both horn, but there is an important difference in that the single unnegated atomic formula in the bodyof the horn clause in the second case involves ~ and in the first it doesn't.This has the effect that any function gives rise to a closure operation onequivalence relations: if / is a binary function and ~ likens a to a' and bto b' then it should also liken f(a,b) to f(a',b').

3 .2 E x t e n d i n g t h e l a n g u a g eSo we have an equivalence relation, written '~' , which is a congruencerelation for various predicates P i: P2 . . . where P is of arity m.Let us now devise a new language, C*. C* will have a new class ofvariables which will be lower case Greek letters: a, ot\, a.2, / ? , /?i, @2 corresponding canonically to the Roman letters a, b of , so that there isa function T that sends Greek letters a,ai, 0:2, . /3,/3i, /?2 to Romanlet ters a ,a 1 ; 02 . . . , b,b\,62,..., c, c i . . . , . . . with which we bega n. We dothis by sending each Greek variable to the corresponding Roman variablewith th e sam e sub scrip t. T he new (Greek) variables in * w ill be usedso that they appear to range over ~-equivalence classes but in fact areontologically cost-free.1

For each old such predicate letter Pi (with ri i arguments) we supplyC with a new predicate let ter P*, also with n ; argu m en ts. T he differencebetween Pi and the corresponding P* has ghosts in na tur al languages. Takefor example the binary relation "fewer than". We say "Five smoked salmonis fewer th a n te n smok ed salmo n" b u t "five is less th a n te n" (no t: "five isfewer th an ten ") . "The tem pe ratu re is ho tter today" sound s wrong: th e dayi s hot ter : the temperature is higher. Function letters are treated similarly.We will have an interpretation T that sends every Greek variable to theRoman variable with the same subscript , and send the predicate let ter P*to Pi.

x I n From a Logical Point of View p 117 (sec t ion 4) Quine makes the poin t tha t i ti s a lways possib le in c i rcumstances l ike th is to quant i fy (apparent ly) over equiva lencec lasses by re in te rpre t ing the congruence re la t ion as '= ' . This may exc i te objec t ions onthe g roun ds tha t ' = ' is no t an o rd in a ry p red ica te to be re in te rp re ted ad iib i tum bu t i spa r t of th e logical vocab ulary . I t i s out of respec t for th i s objec t ion t h a t w e use th espec ial word ' s imu la t io ns ' for those in t e rp re ta t a t io ns th a t t r e a t ' = ' a s a p red ica te l e t t e rinstead of as a par t of the logica l vocabulary .


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A picture might help.

Tvirtual enti t ies * real e nti t iesThe top row contains syntax, and the bottom row contains things. Thevert ically down ward arrows are interp retat ion s as in m odel theory. T he

left-to-right arrow on the top line is an interpretation in the sense of thistra ct . Th e right-to-left arrow on the bottom l ine is the quotient m ap .The point is that composition of arrows make sense: the function X canbe composed with the interpretations and assignment functions used inthe semantics of to give an exactly analogous semantics for *. . .andthe diagram commutes . Virtualism allows one to think that the semanticsarises from the dotted arrows whereas really it arises from the solid arrows.This is pure and simple, and easy to understand, and so a good place tost ar t from. However it will usu ally be insufficient beca use we will wa nt totalk about the new enti t ies and the old in the same breath. What we real lyw ant is a m ap from u * into . It will tu rn ou t tha t the effort involved indoing this gets us also a m ap from a language sl ightly larger than u * . Wecan even introduce at no cost a binary relation symbol and a function symbol. The binary relation symbol is B which has a Ro m an variable to its leftand a Greek variable to its right: a B (3 means tha t a belongs to ( theas i twereequivalence class) /?. The function symbol will be written "a = [6]~"and read "a is the ^-equivalence class of b". Evidently [a]~ is a singular

term in the obvious sense: it 's an abbreviation of '(v/3)(a B /?)', and it islegit imate because 1 of (30)(a B (3) A (V/?')(a B p -+ /3 = /?')) ' is (36)(o ~b A (V6')(a ~ b' > b ~ b') which follows from ~ being an equivalence relation. Finally we have an equality, = . Th is is a two-sorted language, inthat every variable must be of precisely one flavour of the two, Greek andRo m an, and G reek variables cannot be placed as argu m ents to the old predicate letters Pi, nor Roman let ters placed as arguments to the new predicateletters P*. As my late colleague John Iorns once put it "No puns!".

Let us call this new language **, orsince we will not be spendingany time on * and so can recycle its name* again.We now need to show how closed formulae (sentences) in * can betranslated into the old language containing only the old Roman variablesand the original predicates Pi : i I. As before we will use the letter 'X ' forthis interpretation as well as for the map from Greek variables to Romanvariables.


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There is a slight niggle in that we can no longer send a Greek letter tothe Roman let ter with the same subscript . The Roman variable is alreadythe re an d we need the m ap on variables to be injective. W e will dine a tH ilber t 's Hotel. We will send each Greek variable to the correspo ndingRoman variable with the subscript doubled, and we send each Roman variable with subscript n to the Roman variable with the same body and withsubscript 2n + 1.D e f in i t i on 3 . 2 T h e c an on ic a l s im u la t io nDefine X : * > by recursion as follows.(1) X of '=' between Greek variables is '~' ;(2) X of '= ' between Roman variables is '= ' ;(3 ) Xof r P * ~ l i s r P i ~ ' ;(4 ) X sends Roman predicate let ters to themselves;(5) I of a Greek variable is the corresp ond ing Ro m an variable with thesubscript doubled;(6) I of a Ro m an variable with subscript n is the Roman variable with thesame body and subscript 2n + 1;(7 ) X of ra t B ftp is r a 2 i+ i ~ fop;(8 ) X of rajj = [bj}^n is to be ra^i ~ foj+p;(9) X sends quantifiers to themselves.

If T is a theory in then clearly I _ 1 " T is a theo ry in *, and isa conservative extension of T. If we already have a natural deductionformulation for T we can extend i t to a natural deduction formulation forthe corresponding theory X~ l "T in by adding, for each new predicateP * , a pair of a P/ ' - introduction and a P*-elimination rule:Pi(ai ...an) P * ( a i ...an)P*{ai...an) P j ( o i . . . a n )

If we do this we will find that X, considered as a map defined on *-proofs, takes values amon g -pro ofs. T he above rules give rise to derivedrulesJ ( $ ) $

$ J ( $ )There are obvious generalisations of this to the cases where * is many-

sorted, and there is more than one congruence relation in play, one for eachsort. While we are about it , let 's nail down a name for this phenomenon ofhaving congruence relations for each sort.D e f in i t i on 3.3 Let 0 be a pre dica te letter with free variables x where ,for each i I, Xi is of ty pe ov For each i I let ~ , be an equivalence


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relation on things of type a t. If (Vx)(Vy)(((x) A A ^ ~ j yi) > ijj)) thenie iwe say that the bundle {~^: i G / } is a su i t e - o f - c ongruence - r e l a t i on sfor


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there is a canonical simulation, as we have seen. Where there is a canonicalsimulation, i t is natural to think about implementations that respect i t inthe following sense. Recall that an interpretation X is an implementationiff X of equa lity is equality. A little thou gh t will reveal th at it can sendequ ality to equ ality only if it it inc orp ora tes a to ta l function / satisfying(\/xy)(x ~ y f(x) = f(y)), and interprets quantifiers ( 3 O J J ) ( . . . ) as(32i G fuV){---)- {i- e-i quantifiers over Greek variables arise only fromquantifiers over Roman variables that are restricted to the range of /.)

Typically we might also want a sort of representation condition, namely(Vx)(a; ~ /(x)). This might appear to be an obvious absolute requirement,but this turns out not to be so. It seems natural to speak of f{x) as therepresentative of , but if the representation condition is not met, perhaps itwould be better to say f{x) is the prefect of x ( 'prefect' as in 'prefecture'prefects are not elected by the inhabitants of the prefecture from amongtheir number, but are appointed by the central government) .

We say that an implementation X is faithful iff it satisfies this repre-sen tation cond ition. T h at is to say, if it picks a represen tative from eachequivalence class.A weaker condition that is often adequate to our purposes is one

that says that for each predicate P for which ~ is a congruence condi-t ion we have another predicate P' such that (Vxi . . . xn)(P(x\ ... xn) P'(f(x\)... f(xn))). (In realistic cases like the one to follow the map thatsends P to P' is primitive recursive in the Godel numbers.)An illustration of the dispensibility of the representation condition is theVon Ne um ann im plem entatio n of ordin al universally used in wellfounded settheory. A Von Neumann ordinal is not a wellordering, even though it has awellordering indeed a canonical wellordering. Althou gh th e Von Neum annimplementation of ordinals in ZF does not satisfy the representation condi-

tion and therefore is not faithful, it does satisfy the weaker condition of thelast pa rag rap h, a nd it would satisfy t he strong er one if we tweaked it slightlyto take an ordinal to be not a transitive set wellordered by but ratherthe restriction of G to such a set. 2 This is in contrast to the Von Neumannimplementation of cardinals in ZFC, where the cardinal a really is a set ofsize Q , so th e imp lem enta tion really is faithful. In fact th e Von N eum annimplementation of cardinals in ZFC does more still . For cardinals a < (3 wenot only have an injection from f(a) into / ( / ? ) , the injection is actually theinclusion map: (Va/?)(a < (3 > / ( a ) C /( /3)) . (Remem ber tha t C is notthe relation between sets from which < on cardinals arises: < arises from

2 Actually if we take a wellordering to be not a set of ordered pairs but the set ofdomains of i t s in i t ia l segments then the wel lorder ing of a Von Neumann ordina l turnsou t to be the von Neumann o rd ina l itself, so on tha t v iew of wel lorder ing the VonNeumann ordinals are fai thful after al l! We will exploit this fact on p. 67.


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34 Virtual Objects

the plumped out (see page 28) version: {(x,y) : (3x')(x ~ x' C y)} whichis the same as {(x,y) : (3y')(y ~ y' D x)}. (Another fact that dis tractedme for a while!)) . The existence of an implementation satisfying this extra condition is a nontrivial consequence of the axiom of choice. If everyset of cardinals has an order-preserving set of representatives in this waythe n every Dedekind-infmite set has a cou ntably infinite sub set. Howeverit is a theorem of Truss [1973] that every finite set of cardinals admits anorder-preserving set of representatives in this way.

We have said noth ing a bo ut w hethe r the gra ph of / is to be a set, locallya set or even perhaps setlike (a property we will see in definition 5.5 whichhas the effect that if a; is a set of sets, then the collection of cardinals ofmembers of x is also a set) or none of the above. This is being mentionedat this stage merely to reassure the reader thatalthough it is being leftopen it is not being forgotten.3.2.1.1 SubversionImplementations bring with them the possibility of puns. This is becausethe type distinctions that separate Greek from Roman variables can besubverted by the implem entation. Any occurrence of a Greek let ter 'Q J 'in a place where a Roman letter is required can be replaced by ' / ( a * ) ' .Thus it becomes possible to ask questions that sound like "3 5?", namely/ ( 3 ) / ( 5 ) ?Subversion of this kind is well-known in Quin e's NF . Norm ally the axiomof separation says only that if A is a set so is A n {x : (j)} as long as isstratified. Th is restriction seems to keep the system consistent. However,when A is small in a suitable sense ("strongly cantorian") this syntacticrestriction can be subverted in a manner strikingly like that of the previousparagraph .3 . 2 . 2 An illustration: utilitiesLet us have a simple example: utilities as virtu al objects. Utilities arisefrom preference relations, which are (among other things) transitive andreflexive. Binary relations that are transitive and reflexive are preorder s .Each preorder gives rise to a partial order and an equivalence relation in acanonical way. Let < be a preorder. The two relations

~ = {(X:V ) : x < V < x)and


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are an equivalence relation and a partial order respectively. The domain ofthe partial order is the set of ^-equivalence classes of the original domain.As is customary, I have written '[x]^' for the equivalence class of x under~ , and willas is also custom ary om it the subscript where u nd erstan din gis not threaten ed.

Let us now define languages C and *. C contains lower case Romanlette rs for variable s, ' = ' for equality, a bin ary pre dic ate letter < a nd finally'~ ' a binary predicate let ter that will be interpreted by an equivalencerelation. .* contains in addition lower case Greek letters and a binarypredicate letter


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3.2.2.1 Baskets of goodsOf course the domains of realistic preference relations typically have a lotmore s t ructure than th is . A good ( type) 3 need not be merely a thing likesaythe type of apples (one token of which is an apple) but could also bea thing liketo pull a notation out of the airapple apple, a token ofwhich is two apples. A basket of goods is also a good. We would naturallyexpect to be monotone. That is to say we expect (Vabcd)((a < b A c a this will have a solution in IN,but not otherwise. If x is a solution to a + x = b then, for all c, it will besolution to (a + c) + x = (b + c), so the problem posed by the pair (a, b)is in some sense the same as the problem posed by any pair (a + c,b + c).It 's easy to check that two pairs (a , b) and (m, n) pose the same problem(and have the same solution) iff a + n = b + m, so we can take the solutionto a + x = b to be the set {{m, n) : a + n = b + m). If we do this even

3 Here I mean 'type' as in the type-token distinction.


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Second-Order and Higher-Order Theories 37

for equations a + x b that already have solutions then we have a set ofthings on which we can define addition and subtraction and contains allthe natural numbers, or at least duplicates of them.There are various striking differences between this development and thekind of development that would be involved in a logical-constructions-out-of-sense-data development.The first is that analyses of this kind seem to have no ontological bite:there is no suggestion that the construction above shows that integers arenot real objects even though the naturals are, or that i t is possible to do allthe mathematics that needs integers while believing only in IN. Anybodywho has got that far is already up to their navel in abstract entit ies andhas no virtu e left to prote ct. Th ere is noth ing at stake. The purpose ofconstructions like these is to present the new mathematical entit ies in away that explains why we need to believe in them.

Another difference is that with these analyses that are conducted entirely inside the mathematical context, there is usually no question markover the existence of the equivalence classes (and therefore the quotients)modulo the congruence relations. (There was a t ime when there were people who thought that perhaps the collection of all realsor even perhapsthe collection of all naturalswas not a set (which would have had theeffect that sethood of the equivalence classes would have been very hard toprove) but these concerns have now evaporated, and were in any case, notreally concerns about set existence at all but about something much lesscut-and-dried.)

Finally in cases like the construction of the integers from the naturalsthat we have just seen, nobody in their right mind would wish to have anyimp lem entation of integer arithm etic back into the theo ry of ordered p airs ofna tu ra ls th at w as faithful. If one wished to be con crete abo ut it, one wouldidentify the integer n with the set of ordered pairs {{x, y) : x = y + n}, butcertainly not with any one ordered pair in i t . To do th at would be somehowto regard one of the equivalent equations as being more important than theothers, and none of them are.

A development like this one allows what model theorists call eliminationof imaginaries. See Hodges [1993] pp 157ff.

3 . 3 S eco n d - o r d er a n d h ig h er - o rd er t h eo r ie sT he first-order virtua l language arose as a way of conveniently no tatin g factsabout relations between things for which the equivalence relation ~ was acongruence relation. The time has now come to think about a language inwhich we can express facts about sets of things for which the equivalence


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relation in hand is a congruence relation in the appropriate sense. Just aswe did in the first-order case, we will develop this machinery purely for thevirtual entities in the first instance.The higher-order language is of course two-sorted, so we will need theapparatus of suites of congruence relations of definition 3.3Let us suppose that C had upper-case Roman letters for ranging oversets of whatever i t was that the lower-case Roman letters ranged over. C*will of course have upper-case Greek letters to correspond to them. also had G, which had a lower-case Roman letter to its left and an uppercase Roman letter to i ts right. C* will have a corresponding membershiprelation, which we will write 'Gi ' and an equality predicate =1 which sits

between two uppe r-case G reek letters. Gi is of course to be extensional (i t 'sset membership after all) and we know how to interpret in C the equalitybetween lower-case Greek letters. We must have( V e , A)(G =1 A


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Second-Order and Higher-Order Theories 39

De f in i t i on 3 . 6

10 Upper-case Greek letters get sent to the corresponding uppercase Roman let ter with the subscr ipt doubled;11 Upper-case Roman let ters with subscr ipt i get sent to the variable withthe same body and the subscr ipt 2% + 1;12 1 of T , = 1 e r is r G 2 i ~ ! T V ;13 I of r a i 6 i r , -n is r ( 3 a 2 i + 1 ~ a 2 i ) (a 2 i+i G G 2 j ) n ;14 Io f r < / ) * ( e ) n i s r < / ) ( f ) n .

We can add clauses for higher-order analogues of B and the squarebracket notation too if we wish.This definition makes distinctions which we will later collapse. It doesnot assume that sets of whatever it is that the Roman variables of theoriginal languge ranges over can be sets of those sam e things . In th efirst special case of interest to uscardinal numbers as virtual entities oversets we find th at sets of sets are of course sets again , so th at the distinc tionbetween lower and upper case Roman variables is not needed. The distinction between lower and upper-case Greek variables will remain, however: aset of cardinals is not a cardinal, except in a pun!So far we have only considered how to interp ret ta lk ab ou t sets of vir tu alentities, not multisets or lists. No doubt something general can be saidabout relations in the base language from which talk about multisets canarise, but in the only case that we really are compelled to treat (multisetsof cardinals see page 45) talk of multisets arises in a rather pleasing andneat way which would certainly be used to override anything more generalone could do here.

3 . 3 . 1 Third- and higher-order virtual entitiesThis process can be extended any finite number of times we like, so wem ay as well sketc h th e result of doing it infinitely often. It 's qui te h ar dto do this explicitly, as there is no orthographically natural second step ofthe process that took us in one step from lower-case Roman letters (forfirst-order variables) to capital Roman letters (for second-order variables).W h at are we to use for third -ord er variables? Sho rtage of alp ha be ts an dfonts is the single biggest obstacle to understanding and presenting thismater ia l .

Suppose contains variables of all f inite levels too, and a single polymorphic membership predica te le t te r tha t s i t s be tween nt h order and n + l t horder variables for each n.


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Let the higher-order language C* contain, for each n IN, a rangeof variables of level n. C* will also contain predicate letters = n and Ganalogous to =1 and Gi, with the obvious axioms of extensionality andlogical equality axioms.

X of = will be ~ n , which is defined by recu rsion as follows. ~ i ha sjust been defined, and we set ~ + i to be ( ~ ) + .X of Gn+i will be the relation (3a;' ~ n x){x' G Y)And now, as before, we can add to C* new predicate letters corresponding to old predicate letters in for which the equivalence relations{~n: n G IN} are a suite of congru ence relatio ns.Adding more levels (as we are doing here) doesn't commit us to making

m ore copies (as in the discussion on page 32). We still have only two.Later we will be considering constructions that require us to make multiplecopies.


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C h a p t e r 4Card ina l Ar i thmet ic

Re m arkab ly one will no t find a definition of card inal arith m etic in the placeswhere one might expect to find it , such as books or survey articles on thesu bje ct (Holtz-Steffens-W eitz [1999] for ex am ple , or Shelah [1992]) so th er eis still scope for a definition to be hawked. Here is a sloganising suggestion:C a rd in a l a r i t h m et i c is t h e s t u d y o f t h o se re la t io n s b e t w ee nse t s f o r w h ich eq u in u mero s i t y i s a co n g ru en ce re la t io n .

. . . but of course i t 's not quite that simple, because we are going to treathigher-order cardinal ar i thmetic in this chapter , and the subject matter ofhigher-order cardinal arithmetic is that part of set theory for which theequivalence relations of equipollence, equipollence"1" e t c . . . form a suite ofcongruence relat ions.In the terminology introduced earlier, this is virtualism about cardinals.I am setting out not to defend or attack this doctrine but to explain i t : i tneeds to be explained before either of those thing s th an h ap pe n. Virtu alismabout cardinals is a prima facie attractive theory because i t so strikinglyrespects our intuitions that puns l ike '3 6 5' should be accorded groansrather than truth-values. Any account of cardinals that professes to have agood story about 3 e 5 and then tel ls a similar story about commutat ivi tyof addition probably has a suspect account of commutativity of addition aswell. Any re ad er wh o feels th a t th e significance of this p oint is slipping fromtheir gras p need only perform the following thou ght-e xpe rim ent. H ard y

said of Ramanujan that "the natural numbers were his personal fr iends".Now imagine what Ramanujan would reply if one were to say to him "I 'vejust made an exciting discovery that will be very useful to you: a naturalnumber is a wellfounded hereditarily transitive finite set wellordered by G".In the virtualist account cardinal arithmetic is going to arise from settheory, so let 's start with that.

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