Non-Standard Models of Arithmetic

133 Pages·2010·0.74 MB·English

Non-Standard Models of Arithmetic:
a Philosophical and Historical perspective
MScThesis(Afstudeerscriptie)
writtenby
Nicola DiGiorgio
(bornMarch18th,1985inMarianoComense,Italy)
underthesupervisionofDrCatarinaDutilhNovaesandProfDrDickdeJongh,and
submittedtotheBoardofExaminersinpartialfulfillmentoftherequirementsforthe
degreeof
MScinLogic
attheUniversiteitvanAmsterdam.
Dateofthepublicdefense: MembersoftheThesisCommittee:
September3,2010 DrCatarinaDutilhNovaes
ProfDrDickdeJongh
DrBenediktLo¨we
DrKatrinSchulz
Contents
1 DescriptiveuseoflogicandIntendedmodels 1
1.1 Standardmodelsofarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AxiomaticsandFormaltheories . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Hintikkaandthetwousesoflogicinmathematics . . . . . . . . . . . . . . 5
1.4 Typesofcompletenessinformaltheories . . . . . . . . . . . . . . . . . . . 6
1.5 Aheuristictoadescriptiveuseoflogic . . . . . . . . . . . . . . . . . . . . . 9
1.6 FromStandardmodelstoIntendedmodels . . . . . . . . . . . . . . . . . . 11
1.6.1 Intendedmodelsandtheaxiomatizationofarithmetic . . . . . . . 11
1.6.2 Finalremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Non-Standardmodels: anattempteddefinition . . . . . . . . . . . . . . . . 17
2 Thephilosophicalphase-
DedekindandthebirthofNMoA 19
2.1 Dedekindandthebirthofnon-standardmodels . . . . . . . . . . . . . . . 22
2.1.1 Dedekind’sworkontheseriesofnaturalnumbers . . . . . . . . . . 22
2.2 LettertoKeferstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Dedekind’sintroductoryremarks . . . . . . . . . . . . . . . . . . . 25
2.2.2 Dedekind’stenetsfortheseriesofnaturalnumbers . . . . . . . . . 29
2.3 Dedekind’ssixthprinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Theexistenceofnon-standardmodelsofarithmetic . . . . . . . . . 35
2.3.2 AlienintrudersandChains . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 HaoWang: DedekindandNon-standardmodels . . . . . . . . . . . . . . . 44
2.5 Meta-propertiesandArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 Dedekind: completenessandcategoricity . . . . . . . . . . . . . . . . . . . 51
2.6.1 Completecharacterisation . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.2 CompletecharacterisationasCategoricity . . . . . . . . . . . . . . . 52
2.7 Tosumup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
i
CONTENTS
3 Thephilosophicalphase-
SkolemandtheproofoftheexistenceofNMoA 60
3.1 Skolemandnon-standardmodels. . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 SkolemandtheexistenceofNMoA . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 CompactnessandthemodernproofoftheexistenceofNMoA . . . . . . . 68
3.5 Skolem’sproofoftheexistenceofNMoA . . . . . . . . . . . . . . . . . . . 72
3.6 Skolem’sphilosophyandtheproofoftheexistenceofNMoA . . . . . . . 80
3.7 Descriptivevsdeductiveuse: 1stvs2ndorderlogic . . . . . . . . . . . . . 85
3.7.1 Twologicalframeworks . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7.2 Dedekind,Skolemandtheirfoundationalgoals . . . . . . . . . . . 87
4 Themathematicalphaseandtheattemptofaphilosophicalrevival 91
4.1 Anexegeticalandhistoricalclarification . . . . . . . . . . . . . . . . . . . . 91
4.1.1 Theoriginofthephrase”non-standardmodel” . . . . . . . . . . . 91
4.1.2 Doesnon-standardamounttounintended? . . . . . . . . . . . . . . 95
4.2 Themathematicalphase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.1 Fromthephilosophicalphasetothemathematicalone . . . . . . . 98
4.2.2 HenkinandtheordertypeofNMoA . . . . . . . . . . . . . . . . . 99
4.2.3 Twomathematicalresults . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3 Aphilosophicalrevival? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.1 Tennenbaum’sTheoremandStructuralism . . . . . . . . . . . . . . 102
A FirstorderandSecondorderlanguages 110
A.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
ii
Clarityisthegoodfaithofphilosophers.
(ArthurSchopenhauer)
BernardofChartresusedtosaythatwearelikedwarfsontheshouldersofgiants,
sothatwecanseemorethanthey,andthingsatagreaterdistance,
notbyvirtueofanysharpnessofsightonourpart,oranyphysicaldistinction,
butbecausewearecarriedhighandraisedupbytheirgiantsize.
(JohnofSalisbury)
Alla mia nazione
Nonpopoloarabo,nonpopolobalcanico,nonpopoloantico
manazionevivente,manazioneeuropea:
ecosasei? Terradiinfanti,affamati,corrotti,
governantiimpiegatidiagrari,prefetticodini,
avvocatucciuntidibrillantinaeipiedisporchi,
funzionariliberalicarognecomegliziibigotti,
unacaserma,unseminario,unaspiaggialibera,uncasino!
Milionidipiccoliborghesicomemilionidiporci
pascolanosospingendosisottogliillesipalazzotti,
tracasecolonialiscrostateormaicomechiese.
Proprioperche´ tuseiesistita,oranonesisti,
proprioperche´ fosticosciente,seiincosciente.
Esoloperche´ seicattolica,nonpuoipensare
cheiltuomalee´ tuttomale: colpadiognimale.
Sprofondainquestotuobelmare,liberailmondo.
(PierPaoloPasolini)
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Abstract
Overthelastfiftyyears, thestudyofnon-standardmodelsofarithmetichasbecome
a fertile and highly technical mathematical branch. Nevertheless, surprising as it might
seemtoday,thetopicofnon-standardmodelswasbornasagenuinephilosophicalissue.
My work sets out to investigate the philosophical and historical significance of non-
standardmodelsofarithmetic. Notonlyhasnon-standardmodelsofarithmeticencoun-
tered a scarce success in what concerns philosophy, but also a detailed presentation on
thebeginningsofthetopichasneverbeenwritten.
Inparticular, Iwilldiscusstheoriginsofnon-standardmodelsofarithmeticwithre-
specttothepioneeringworkofRichardDedekindandThoralfSkolem,andstrivetoshed
light on the relationship between philosophy, logic and mathematics in what concerns
arithmeticaltheoriesandfoundationalstudiesinmathematics.
Keywords: Non-standardmodels, intendedmodel, descriptiveuseoflogic, Peano
Arithmetic,Dedekind,Skolem,Tennenbaum’sTheorem,foundationsofmathemat-
ics.
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Introduction
To fully grasp the significance of NON-STANDARD MODELS, we have to go back to the
originsofmathematicallogic. Thatistosay,totracethehistoricaldevelopmentsthathave
resulted in what we call logic today. Dry as it may seem at a first glance, the historical
approachIpursuehereislikelytocomeoutasanexcellentwaytocastafresh,brand-new
lightonage-oldissuesthatseemedtobesettledonceandforall. Aswegraduallymove
towards the role of non-standard models of arithmetic, history, philosophy, logic and
mathematicscomeoutasinseparablyinterwoven. Ontopofthat,notonlyhasthetopic
of NON-STANDARD MODELS encountered a scarce success in what concerns philosophy,
butalsoaphilosophicalandhistoricalpresentationaboutthebeginningsofthetopichas
never been written. Hopefully, this research may constitute a starting point for much
neededfurtherworkonthesubject.
WhyNon-Standardmodels?
Today the issue of NON-STANDARD MODELS represents a highly technical mathemat-
ical field. However, to tell the story of these models, we have to turn to philosophy
prior to mathematics. Surprising as it may seem today, I argue that the mathematical
researchconcerningnon-standardmodelsisgroundedonprofoundphilosophicalroots.
Thisworkintendstoinvestigatetheseroots,whichmayultimatelyshednewlightonthe
recentmathematicalresults.
WhyNon-Standardmodelsofarithmetic?
Needless to say that the phrase “non-standard models” does not pertain to the sole
arithmetical sphere, it may also refer to the set-theoretical case (pointed out by Skolem in
1922astheexistenceof“countable”modelsofthefirstorderaxiomatisationofset-theory
—aphenomenonknowntodayasSkolem’sParadox),ortonon-standardanalysis(which
dealswiththeexistenceof“special”numbers,callednon-standard,exploitedtoprovide
arigorousdefinitionofelementsknowninanalysisas“infinitesimals”).
Nevertheless, inthisworkIwillconfinemyselftothearithmeticalcase. Thereasons
aremainlythree:
• Ibelievethatarithmeticrepresents,inthisrespect,thefirststepintheinvestigation
v
of non-standard models. In fact, the meagre presence, in the philosophical litera-
ture,ofauniformandall-comprehensivetreatmentofnon-standardmodelsmakes
thearithmeticalaccountaseminalandself-containedresearch;
• forahistoricalandexegeticalreasons: thelargenumberofhistoricalpapersinarith-
metic lends itself to an in-depth study that can represent a good starting point for
new perspectives on old debates (e.g. whether arithmetical theories could reach
somedegreeofcompleteness;whichlogicisthemostsuitableforarithmeticalthe-
ories;etc);
• fortherelevancewithrespecttothepresent-daydebateinphilosophyofmathemat-
ics: thefieldofnon-standardmodelscanprovidesomeresults(e.g.Tennenbaum’s
Theorem)forthecurrentphilosophicaldiscussion.
To sum up, in this thesis I will emphasise the importance of non-standard models of
arithmeticasagenuinephilosophicalissue.
Thephilosophicalandmathematicalphaseinthedevelopmentofnon-standardmod-
elsofarithmetic
Commonly,thespacededicatedinlogicaltextbooks,ifany,tonon-standardmodelsis
quite exiguous: Hodges discusses them in half a page; Van Dalen does not write more
than a couple of pages on them; the only exception is Boolos’ book where a seventeen
pagechapterisdevotedtoarathertechnicalpresentationofthetopic.1
On the other hand, there are only two fundamental references on the non-standard
models,so-called”bibles”bytheinsiders,thatdojusticetowhatshouldberegarded,by
thistime,asanimportantandpromisingfieldofresearch: [kaye91]andthemorerecent
[kossak06].2 On top of that, we should note that these two monographs are strongly
mathematicallyoriented.
At the end of the day, it would then seem that the concept of non-standard models
haslittletocontributetophilosophicaldebates. Roughlyspeaking,thetwomathematical
studies along with the meagre logical treatment exhaust the literature on non-standard
models.
An exception is [smorynski84] which was indeed a forerunner attempt to give a
historical account of non-standard models. Before the publishing of [kaye91] in 1991,
”the only, and not easily available, source was the excellent notes of Craig Smorynski
[smorynski84] from his lectures on nonstandard models at the University of Utrecht in
1978.”3
1See[hodges01,p.70],[vandalen04,p.113,pp.121-122],[boolos07,ch.25]
2Strictlyspeaking,[kossak06]presentsitselfascontinuationof[kaye91].
3[kossak06,p.viii]
vi
Although Smorynski’s lecture notes have the purpose of presenting the main math-
ematical results, at the beginning of his text some historical considerations are put for-
ward. Accordingtohim, theintentionofthenotesis”togiveapartlyhistoricalaccount
ofthedevelopmentofthesubject.”4Inparticular,Smorynskipointsoutthattwodifferent
approachesofnon-standardmodelscanbeidentified.
It is also worth mentioning that Skolem’s goal5 in constructing nonstandard
models was philosophical: He aimed to shew that first-order logic could not
characterise the number series; he did not care to start a new subject. Until
the1960s,thiswasgenerallythecase-nonstandardmodelsofarithmeticwere
either objects of philosophical interest or tools, not objects of mathematical
interestintheirownright. [smorynski84,p.3,myemphasis]
For Smorynski, Skolem considered non-standard models as nothing but limitative
results, whereas from the 1960s onwards these models gained their own importance as
entities.
Following Smorynski, we distinguish two historical phases: one ”philosophical”,
which is characterised by the work of Dedekind (as will argue) and Skolem, and the
other one ”mathematical” which goes from the 1960s and onwards. The quotation of
Smorynski appears in the first paragraph that he entitles ”The Beginnings (the 1950s
and Earlier)”. Nevertheless, Smorynski does not elaborate further on this ”philosoph-
icalphase”inthelecturenotes,heratherfocusesmainlyonthemathematicalresultsthat
havebuiltupnon-standardmodelsasamathematicalfieldinitsownright.
The thesis strives to fill such a historical gap and provides a better picture of the
philosophicalrootsofnon-standardmodelsofarithmetic,andtheirinfluenceonthenext
mathematicaldevelopments.
Todoso,wewilldiscussindetailontheimportanceofthephilosophicalphasebypre-
sentingitsfeatureswithrespecttothemathematicalphase. WethenarguethatDedekind
has to be considered, together with Skolem, as the forefather the non-standard models
andthusviewedasamemberofthephilosophicalphase.
Inparticular,wewillconsiderDedekind’slettertoKefersteinandhiscategoricalchar-
acterisationofhissystem. Afterthat,wewilldiscussSkolem’sproofsoftheexistenceof
non-standardmodelsandtheirrelationshipwithSkolem’sfinitistmathematicalcredo.
Moreover, we will also argue for a possible third ”hybrid” phase which could be
the attempt to recover from the mismatch between the scarce interest that philosophy
haspaidtonon-standardmodelsoverthelast60years, andtheabundantmathematical
resultsfromthe1960’son. Thisphasepossiblysetsoutfromthephilosophicalrelevance
ofresultsconcerningnon-standardtodebatesinphilosophyofmathematics.
4[smorynski84,p.1]
5NotethatThoralfSkolemiscommonlycreditedwithbeingtheonewhodiscoverednon-standardmod-
els.However,wewillarguethat,inthisrespect,Dedekindshouldbementionedalso.
vii
Overviewofthethesis
In chapter 1 we present some introductory remarks about important distinctions that
aredealtwithinthetext—suchasanalytic/syntheticreasoning,deductive/descriptive
use of logic in mathematics, intended/unintended models — in order to put forth their
roleinthediscussionofnon-standardmodelsofarithmetic(NMoAforshort).
Inchapter2weconsiderDedekind’sworkwithrespecttoNMoA,especiallyhis”The
Nature and the Meaning of Numbers” and one of the letter he addressed to Keferstein.
We argue that in the letter to Keferstein we find the very first evidence of the existence
of NMoA. Moreover, we illustrate Dedekind’s attempt to banish non-standard models
inhisaxiomatizationofarithmeticviaemployingstatementsexpressibleinsecond-order
logic.
Inchapter3weexploreSkolem’soriginalproofoftheexistenceofNMoA,knownin
the literature as ”ultrapower-like” construction, the impact that Skolem’s philosophical
viewshadontheproofandthewaynon-standardmodelswerereceivedfromfollowing
philosophers.
In chapter 4 we outline the main features of the succeeding mathematical phase.
In particular we focus on one mathematical result, viz. Tennenbaum’s Theorem, which
could be the sparkle of the return of philosophical interest to the issue of non-standard
models.
In the appendix we provide the syntax and the semantics of first and second order
logic.
viii
Acknowledgements
This thesis would have never been conceived without the supervision of dr. Dutilh No-
vaes. ShesuggestedmetheproblemthatIinvestigateinthisthesis,i.e.thephilosophical
significance of the non-standard models of arithmetic. I would like to express my grati-
tudetoherfortheencouragement,suggestions,andcriticism.
I am also indebted to prof. Dick de Jongh who has done a very careful work of revi-
siontothepreviousversionsofthisthesis. Muchoftheclarityofthepresentredactionis
duetohissuggestions.
Ialsowouldliketothankprof. SergioGalvanwhogavemethefirstcoursesinlogic,
encouragedandhelpedmetogoabroadandcompletemyMaster,andprof. Alessandro
GiordaniwhofirstintroducedmetothestudyoftheworkofRichardDedekind.
Inoltre, vorrei ringraziare con grande affetto i miei genitori, Tina e Guido, e la mia
caraSofietwzka,chemihannosempresupportatoe”sopportato”inquestianni.
UnringraziamentospecialevaancheaPasqualina(dettaSheiluzziella),eatuttal’allegra
brigata composta da Giannino, (ovvero Gianni o GianPorco), Cicciuzzo (detto anche il
Commendatore),ilmagoPotter,Edo(ovveroPierre,dettoancheilDrago),BobGiovaDy-
lan,FrancyRaiworld,Brennon-Brendon(dettoildottore),Micky-Smicky,LucaSambuca,
SilviaBerluscona,EbevilOtutto,etuttiquellichemihannoaccompagnatoinquestidue
anni.
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