Finite Type Arithmetic

123 Pages·2010·0.77 MB·English

Finite Type Arithmetic
Computable Existence Analysed by Modified Realisability and
Functional Interpretation
H
KlausFrovinJørgensen
Master’sThesis,March19,2001
Supervisors: UlrichKohlenbachandStigAndurPedersen
DepartmentsofMathematicsandPhilosohy
UniversityofRoskilde
Abstract in English and Danish
MotivatedbyDavidHilbert’sprogramandphilosophyofmathematicswegiveinthecontext
ofnaturaldeductionanintroductiontotheDialecticainterpretationandcomparetheinterpre-
tationwithmodifiedrealisability.Weshowhowtheinterpretationsrepresenttwostructurally
differentmethodsforunwindingcomputableinformationfromproofswhichmayusecertain
primafacie non-constructive(ideal)elementsofmathematics. Consequently,thetwo inter-
pretationsalsorepresentdifferentviewsonwhatistoberegardedasconstructiverelativeto
arithmetic. Thedifferencesshowupin theinterpretationsofextensionality,Markov’sprin-
cipleandrestrictedformsofindependence-of-premise.Weshowthatitiscomputationallya
subtleissue tocombinetheseidealelementsandprovethatMarkov’sprincipleiscomputa-
tionallyincompatiblewithindependence-of-premisefornegatedpurelyuniversalformulas.
In the contextof extracting computationalcontentfrom proofs in typed classical arith-
meticwealsocompareinanextensionalcontext(i)themethodprovidedbynegativetrans-
lation + Dialectica interpretation with (ii) the method provided by negative translation +
w
A-translation+modifiedrealisability.NoneofthesemethodscanbeappliedfullytoE-PA ,
w
sinceE-HA isnotclosedunderMarkov’srule,whereasthemethodbasedontheDialectica
interpretationcanbeusedifonlyweakextensionalityisrequired.
Finally, we presenta new variantof the Dialectica interpretationin orderto obtain(the
well-known)existenceproperty,disjunctionpropertyandotherclosureresultsfortypedin-
tuitionisticarithmeticandextensionshereof. Henceitisshownthatfunctionalinterpretation
canbeusedalsoforthispurpose.
◭◮
Vi giver med udgangspunkt i Hilberts program og hans matematikfilosofi en introduktion
til Dialectica fortolkningeninden for naturlig deduktion. Denne fortolkning sammenlignes
med modificeret realiserbarhed, og vi viser, hvordan de to fortolkninger repræsenterer to
struktureltforskelligemetodertilatuddriveberegnbarinformationfraikke-konstruktivebe-
viser.Sa˚ledesrepræsentererdetofortolkningerforskelligesynpa˚,hvaddererkonstruktivt.
Forskellighedernevisersigisæriforbindelsemedekstensionalitet,Markovsprincipogprin-
cippet om uafhængighed af præmis. Vi viser, at det er et subtilt emne at kombinere disse
ideal-elementer,ogatMarkov’sprincipogprincippetomuafhængighedafnegeretuniverselle
præmisserisærdeleshederuforligneligemedhensyntilberegnbarhed.
Medhensyntilberegnbartindholdafklassiskebeviserindenfortypetaritmetiksammen-
lignerviienekstensionelkontekst(i) metodengivetvednegativoversættelse+ Dialectica
fortolkningmed(ii)metodengivetvednegativoversættelse+A-oversættelse+modificeret
w
realiserbarhed.Ingenafdisse metoderkanbenyttesifuldudstrækningindenforE-PA ,da
w
E-HA ikkeerlukketunderMarkovsregel.Metodenbaseretpa˚Dialecticafortolkningenkan
derimodanvendes,hvisdetkunkræves,atteorienharsvagekstensionalitet.
TilsidstintroduceresennyvariantafDialecticafortolkningen.Dennebenyttestilatvise
(de velkendte) eksistensegenskab, disjunktionsegenskab og andre egenskaber med hensyn
tillukkethedfortypetintuitionistiskaritmetikinklusiveenekstension.Sa˚ledesvisesdet, at
funktionalfortolkningogsa˚ kanbenyttestildetteforma˚l.
iv
Preface
Thismaster’sthesisiswrittenforobtainmentofamaster’sdegreeinbothmathematicsand
philosophy. Hence, the present text treats topics from both fields, as they are connected
in logic. The specific purpose is to give a systematic analysis of certain prima facie non-
constructiveelementsofmathematicsbyusingdifferentinterpretationsandtranslationspro-
videdbyprooftheory. Myhopeisthatthereaderwillacknowledgethistohavesignificance
bothwithinmathematicsandphilosophy.
My gratitude goes to my supervisors: Ulrich Kohlenbachin mathematics and Stig An-
dur Pedersen in philosophy. Ulrich Kohlenbachhas been most generouswith his time and
expertise,andallresultspresentedhereareobtainedincollaborationwith him. Stig Andur
Pedersenintroducedmeseveralyearsagotologicandprooftheoryandtoa veryfruitful,I
think, view on mathematics. They have both in a very friendly and constructive way sup-
portedandguidedme.
IalsowanttothankUlrichBergerforgivingmetheidea,thatoneshouldgiveaDialectica
interpretationwithinnaturaldeduction. Ihavegainedalotinpursuingthis. Withrespectto
thefinalproofreadingIwanttothankVincentF.Hendricksforhisthoroughtreatment;also
Dan Temple adjusted my language. Finally, thanks also to Jesper H. Andersen and Ivar
Rummelhoffforusefulcomments.
v

Contents
1 SettingandObjectofSurvey 1
1.1 Hilbert’sprogramandviewonmathematics . . . . . . . . . . . . . . . . . . 3
1.2 FailureofHilbert’sprogram—failureofseparation . . . . . . . . . . . . . . 5
1.3 TheproofinterpretationbyHeyting . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Go¨del’sviewonaconstructivefoundation . . . . . . . . . . . . . . . . . . . 9
1.5 Motivationforandfocusofthisthesis . . . . . . . . . . . . . . . . . . . . . 11
2 IntroductiontoGo¨del’sDialecticaInterpretation 15
2.1 AHilbertsystemforweaklyextensionalHeytingarithmeticinallfinitetypes 15
w
2.2 AxiomsandrulesofWE-HA . . . . . . . . . . . . . . . . . . . . . . . . . 17
H w
2.3 WE-T asthequantifierfreesubsystemofWE-HA . . . . . . . . . . . . . 21
H H
2.4 DefinitionandanalysisofDialecticatranslation . . . . . . . . . . . . . . . . 22
w
2.5 InterpretationtheoremforWE-HA . . . . . . . . . . . . . . . . . . . . . . 27
H
3 InterpretationTheoremswithinNaturalDeduction 29
w
3.1 FormulationofWE-HA andWE-T . . . . . . . . . . . . . . . . . . . . . 29
ND ND
3.2 Discussionofthedeductiontheorem . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Translationofderivationsunderassumptions . . . . . . . . . . . . . . . . . 36
w
3.4 InterpretationtheoremforWE-HA . . . . . . . . . . . . . . . . . . . . . . 37
ND
3.5 RelationbetweentheDialecticainterpretationandtheDiller-Nahminterpre-
tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 IntuitionisticlinearlogicandtheDialecticainterpretation . . . . . . . . . . . 49
w w w
3.7 InterpretationtheoremforWE-HA +MP +IP +AC . . . . . . . . . . . 51
3.8 Thenon-constructivetheoryWE-HAw +IPw +M∀ Pw . . . . . . . . . . . . . 52
¬∀
4 ClassicalArithmeticandKuroda’sNegativeTranslation 55
w
4.1 FormulationofWE-PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Fromclassicaltointuitionisticlogic:Kuroda’snegativetranslation . . . . . . 56
w
4.3 InterpretationtheoremforWE-PA +QF-ACandconsistencyofthetheory . 59
4.4 Extractiontheoremandconservativeness . . . . . . . . . . . . . . . . . . . . 60
4.5 Thephilosophicalsignificanceofinterpretationtheorems . . . . . . . . . . . 62
w
4.6 PAasasubsystemofWE-PA . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Theno-counterexampleinterpretationofPeanoarithmetic . . . . . . . . . . 65
5 ModifiedRealisability,A-translationandApplications 69
w
5.1 DefinitionofE-HA andmodifiedrealisability . . . . . . . . . . . . . . . . 70
w w
5.2 RealisabilityinterpretationofE-HA +AC+IP . . . . . . . . . . . . . . . 72
ef
5.3 Contractionandnegativelyoccurringuniversalquantifiers . . . . . . . . . . 74
5.4 Markov’sprincipleintuitionisticallyunprovable . . . . . . . . . . . . . . . . 75
w w
5.5 E-HA IP ACnotclosedunderMarkov’srule . . . . . . . . . . . . . . 75
± ef±
vii
viii Contents
5.6 TheFriedman-DragalinA-translationforformulasofHA . . . . . . . . . . . 77
5.7 Realisabilitywithtruth:Closureproperties. . . . . . . . . . . . . . . . . . . 80
6 ClosureunderRulesbyFunctionalInterpretations 84
w
6.1 Thereisno‘Dialectica-with-truth’interpretationofWE-HA . . . . . . . . . 84
6.2 DefinitionandsoundnessofQ-translation . . . . . . . . . . . . . . . . . . . 86
6.3 ClosurepropertiesofintuitionisticarithmeticshowedbyQ-translation . . . . 98
6.4 Q-interpretationisnotclosedunderdeductions . . . . . . . . . . . . . . . . 99
6.5 ClosurepropertiesofWE-HAw +IPw +MPw +AC+G byDialectica . . . . 100
6.6 ConstructivenessofWE-HAw +IPw ∀+MPw +AC+G . . . . . . . . . . . . . 101
∀
7 Conclusions 102
7.1 Interpretingmathematicsbymethodsfromprooftheory . . . . . . . . . . . . 103
7.2 EvaluationofmodifiedrealisabilityandDialecticainterpretation . . . . . . . 105
7.3 Twostrongconstructivetheories . . . . . . . . . . . . . . . . . . . . . . . . 106
7.4 Dialecticaasatoolinproofminingandreductiveprooftheory . . . . . . . . 106
7.5 Differentinterpretations—differentvalidations:Mathematicsandlanguage . 108
Bibliography 110
CHAPTER 1
Setting and Object of Survey
Thisthesisconcernsprooftheoryandthesignificanceofcertainprooftheoreticalinvestiga-
tions.Prooftheoryisthepartofmathematicallogicwhichstudiestheconceptsofmathemat-
ical proofs and mathematical provability. Proofs are an indispensable part of mathematics
andthereforeprooftheoryisalsoastudyofthefoundationsofmathematics.Letuselaborate
ontheconnectionbetweenprooftheoryandphilosophyofmathematics.
First, proof theory studies proofs as formal objects whereas, generally, mathematical
proofsareinformalandtoacertaindegreeimprecise. Thereis,however,inpracticeaclose
affinitybetweenformalandinformalproofs.Partsofproofsusedinmathematicalarguments
canbemoreorlessformal.Butincasethereisadisagreementonaproofonecantrytomake
theproblematicpartoftheproofmoreformalinordertobepreciseonthematter. Onecan,
inotherwords,formaliseindifferentdegreesandthecompleteformalproofcanbeviewedas
somekindofidealisationofthemathematicalproof. Itisinthisrespectprooftheorystudies
mathematicalproofs.
Second, from results obtained in the 1930s in mathematical logic it follows that there
cannotbeformaltheorieswhichrepresentallofmathematics: Foranygivenformaltheory
containinga minimumofarithmetictherewillalwaysbemathematicaltheoremswhichare
independentofthattheory. Thereare, inotherwords,theoremsofthetheorywhichcannot
beprovednordisproved. Itis,nevertheless,possibletorepresentbigbodiesofmathematics
informalframeworksandtostudyinternalrelationships.Apartofmathematicallogiccalled
reversemathematicsisanimpressiveexampleofthis; seee.g.(Simpson;1999). Therefore,
relativetospecificmathematicaltheoriesprooftheoreticalinvestigationscanhaveimportant
philosophicalconsequences.Wewillcallsuchinvestigationslocal.
Butofcourse,prooftheorydoesnotgivethefullpictureofaphilosophyofmathematics.
Generalepistemological, ontological, sociologicaland historicalissues – which are treated
in philosophy,philosophyof science and history of science – play an importantrole in the
overall philosophical analysis of mathematics. In this thesis, however, we will only touch
the epistemological and ontological questions in so far as is necessary in order to derive
philosophicalconsequencesoftheprooftheoreticalinvestigations.
We will apply different methods from proof theory in order to study constructivity of
proofs and principles. Now, it is in itself a difficult question to point out what constitutes
a constructiveproof. Below we will line up certain propertieswhich a constructivesystem
shouldmeet. Butasitturnsout,thereisnounambiguouscharacterisationofconstructivism.
Anyhow,adesirablepropertyofaconstructivetheoryiscertainlythefollowing: Ifwehave
provedanexistentialstatement xA(x)thenwecaninfactexhibitanobjecttsuchthatAholds
∃
fort.1 Similarly,ifwehaveprovedadisjunctionA B,withnofreevariablesthenwecantell
∨
whichofthe two actuallyholds. We willnow givea simpleexampleof a non-constructive
proof,duetoH.Friedman.
1Foraninteresting andinformative survey ofthe historical roots ofthese matters (whichat least goback to
WhiteheadandRussell’sPrincipiaMathematicafrom1910–13)seeMancuso(n.d.). Inconnectionwiththispaper
itshould,however,benotedthatitissomehowdifficulttofollowMancuso’sseparationofdirectandindirectnon-
constructiveproofs,sincetheybothbytheendofthedayrestontertiumnondatur.
1
2 SettingandObjectofSurvey
Theorem.e p isirrationalore+p isirrational.
−
Proof. Assumetheybotharerational. Sincethe sumoftworationalsisa rationalwe have
that2eisrational.Contradiction.
⊣
Itisquestionablehowmuchinformationthisproofcontains. Surely,theproofdoesnot
tell us which of the two disjuncts is irrational. A little surprising due to the simplicity of
theproblem,nomethodorproofonthisproblemhasuntilnowshownuswhichofthe two
actuallyis irrational. Theaboveproofonlytells usthatbothcannotberational, butitdoes
notexhibitanumberofthetwosuchthatthisnumberistheone—clearlyitwouldhavebeen
moresatisfyingifweknewthat.
Therearemanyexamplesfromallovermathematicswhereexistencestatementsordis-
junction statements are proved but the proofs do not provide us with any instances or any
algorithmswhichmakeitpossibleforustoseewhythetheoremsaretrue—allweseeisthat
itcannotbethecasethattheirnegationsaretrue. Thisiscertainlyconstructivelyunsatisfy-
ing.2 Indirectproofsareingeneralnotconstructive. Butinlotsofcasestheyareorseemto
betheonlypossible;asforinstanceintheaboveproof.
At the entrance to the 20th century mathematics had its so-called foundational crisis,
where fundamentalquestions appeared during the developmentof new mathematical theo-
ries. AnexampleofsuchatheoryissettheoryasdevelopedbyGeorgCantor(1845–1918),
whointroducedabeautifulandusefultheoryoftransfinitearithmetic—arithmeticoninfinite
numbers.Anotherexampleisthedevelopmentofageneraltheoryoftopology,mainlybyFe-
lixHausdorff(1868–1942),whereconceptsfromanalysissuchascontinuityweregeneralised
toquiteabstractnotions.Manyofthesenewtheoriesandconceptsinvolvedinfinitetotalities
and, moreover, inconsistencies and difficult open foundationalproblemswere found in the
veryfirstformulationsofsettheory. Theparadoxof BertrandRussell (1872-1970)showed
thatnotallpropertiescanbeusedfordefiningsetsandCantorposedthecontinuumhypoth-
esis (CH) which asserts that there is no set with cardinality strictly greater than the set of
naturalnumbersbut also strictly less than the set of real numbers. If one views the CH as
a definitemathematicalproblemthenit hasremainedopen—ithasevenbeenshownthatit
cannotbedecidedwithintheZFCformalisationofsettheory.
Meanwhile,alsoprooftechniqueswerequestioned.In1888DavidHilbert(1862–1943)
provedhisfamousbasistheoremininvarianttheorybynon-constructivemethodsbutexperts
on the field – e.g. P. Gordan– called Hilbert’s proof “theology”.3 L.E.J. Brouwer (1881 –
1966),thefounderofintuitionism,bannedinhisthesisfrom1907indirectproofsandnon-
constructivemethodsin general. Also HermannWeyl (1885 – 1955), a studentof Hilbert,
becameunsatisfiedwithsettheoreticalfoundationsanddevelopedin(Weyl;1918)hiskind
ofconstructivemathematics,whichlaterbecameknownunderthename“predicativemath-
ematics”. Theso-called“Grundlagenstreit”wasjustabouttostart. Manyimportantmathe-
maticianssuchasHilbert,BrouwerandWeyltookpartin it, and–aswe shallsee– sodid
2Itisalsoepistemologically problematictoassumethatanymathematicalstatementhasatruthvalue—alsoin
caseswhereitcompletelyimpossibleforustogettoknowwhatthatvalueis. Wewillnot,however,dealwiththat
problemhere,butseeforinstance(Jørgensen&Pedersen;2000). Also:Dummett(1977)arguesthattruthmustbe
somethingwhichisaccessibletous.
3Moreonthisin(Rowe;2000).
1.1 Hilbert’sprogramandviewonmathematics 3
KurtGo¨del(1906–1978)andArendHeyting(1898–1980). Nowandthenthediscussions
were rather emotional, but many importantnotions, insights and interpretationsarose from
thisintenseperiod.Oneofthemwasprooftheory.
1.1 Hilbert’sprogramandviewonmathematics
Hilberthadalongmathematicalcareerandwascontributinginnearlyallareasofmathemat-
icssuchasalgebra,geometry,numbertheory,mathematicalphysics,mathematicallogic,etc.
He had worked with the new mathematical theories and seen into the uncountable infinite
thatCantor’ssettheorywasabout. Hehadseenandusedtheeffectivenessandeleganceof
non-constructivemethods.Thus,“AusdemParadies,daßCantorunsgeschaffen,sollunsnie-
mandvertreibenko¨nnen”,asHilbert(1926,170)putsitinhisfamousmetaphor. Therefore
hedevelopedhisprogram.
The program was at heart Kantian.4 It was formulated fully – by Hilbert and his co-
workers, especially PaulBernays(1888–1977)– and putforth in the 1920s(Hilbert;1922,
1926). Thephilosophicalpositionbehindtheprogramisthefollowing. Mathematicscanbe
splitintotwoparts:
1. Thefinitary(orcontentual)partofmathematics.
2. Theidealpartofmathematics.
The finitary part of mathematics was meant to be that part of mathematics of which there
couldbenodoubts:Finitaryreasoningaboutthenaturalnumbers,i.e.nounrestrictedquanti-
fiers,andsimplereasoningonfinitegraphsandgeometricalfigures. Todaythereisageneral
agreementon, thatwhatHilberttookto be the finitary partof (informal)mathematicswas,
atleast, whatcanbe codedandjustified by primitiverecursivearithmetic(PRA), see (Tait;
n.d.)foradiscussionofthis. Idealmathematicswastakentobethehighlyabstractelements
ofmathematicsofwhichtheontologicalstatuswerenotimmediate. Examplesofsuchideal
parts of mathematics could be (i) completed infinities, such as the set of natural numbers
w , needed to develop a theory of the transfinite; (ii) the expansion of the real numbers by
thecomplexnumberi=√ 1,whichenablesustoprovethefundamentaltheoremofalge-
−
bra; (iii) the ‘completion’of Euclidean geometryby an infinite line consisting of points at
infinitythusobtainingprojectivegeometry,andso forth. Assuch, theidealelementscould
not, according to Hilbert, be perceived by the senses. They had more the role of complet-
ingthefinitaryandregulatingit—inthesame wayasinKant’stheoryofknowledgewhere
ideas of reason regulate knowledge.5 The ideal elements were supposed to be abstract el-
ements introduced in the development of mathematics in order to simplify, generalise and
completealreadyexistingmathematics. Butinsuchaprocessnewmathematicswouldalso
arise and this was how Hilbert saw the expansion and progression of mathematics. These
4DetailsonthisisprovidedinAndersen(2000). AndersenalsodiscussestheconflictsbetweenKant’sviewon
mathematicsandHilbert’sviewonmathematics—inparticulartheroleoftheinfinite. Athemewewillnottouch
here,butseealsoMajer(1993),Detlefsen(1995)andPosy(1995).
5Note, that Kant (1781/87) indeed hadtwo kinds of‘ideas’: (i)Theconstitutive useoftranscendental ideas
causing paralogisms and antinomies ofpure reason (Kant; 1781/87, A338-A567); and (ii) the regulative use of
generalideasofpurereason,asusedinscience(Kant;1781/87,A642-A704).
4 SettingandObjectofSurvey
ideal elements of the mathematical method and universe were, of course, of indispensable
value. However,Hilbertwasatthesame timeawareofthefactthatthisprogressivenessof
the mathematicalmethodwas transcendingthe securedfinitary partsof mathematicsandit
wasthereforeinneedofsomekindofjustification.
Thisjustificationwouldconsistinshowing,mathematically,thattheidealpartofmathe-
maticscouldnotprovenewpurelyfinitarystatements,i.e.couldnotprovefinitarystatements
which were not provablealready in the finitary part of mathematics. This is where Hilbert
connectedtheaxiomaticapproach,theideaofa‘prooftheory’,withhisgeneralviewonmath-
ematicsasjustdescribed. Inmoderntechnicaltermsthegoaloftheprogramisdescribedby
thefollowing.
LetSbesomeformalsystemrepresentingmathematics—boththeidealandthe finitary
part.AformulainthelanguageofSisafiniteobjectanditcanthereforebecodedeffectively
by a natural number; proofs in S can likewise be coded. Thus, Proof (x,y) is a predicate
S
obtaining between two natural numbers x and y expressing that x encodes a proof in S of
someformulahavingcodey. Asisstandard,letpAqbethecodeofA. (Theargumentbelow
isinthegivenformalittlevague—thedetailsrelyonthespecificpropertiesoftheencoding;
see (Smorynski;1977,sect. 2–4).) Intechnicaltermsthe essenceofHilbert’sprogramwas
thatforanyfinitarystatementR(x)withxasfreevariablethereflectionprinciple
Proof (u,pR(x˙)q) R(x) (1.1)
S
→
shouldbeprovablebyfinitarymeans(wherex˙referstothex-thnumeral).However,itwould
besufficienttoestablishconsistencyofSinafinitaryway. ForsupposeonehasaproofinS
ofsomefinitarystatementR(x)containingonlyxasfreevariable,hence
Proof (u,pR(x˙)q) (1.2)
S
wouldbefinitarilyprovable. However,ifR(x)werenottrueforallxthenforsomec, R(c)
wouldbeprovablewithinS.Infactwewouldhave,duetoS completeness, ¬
1
R(x) Proof (v ,p R(x˙)q), (1.3)
S x
¬ → ¬
wherev dependsonwhichvaluextakes. If,ontheotherhand,wecouldproveconsistency
x
ofSbyfinitarymeanswewouldhave
Proof (u,pR(x˙)q) Proof (v,p R(x˙)q) . (1.4)
S S
¬ ∧ ¬
(cid:0) (cid:1)
Now,(1.2),(1.3)togetherwith(1.4)impliesthat R(x)hasafinitaryproof,andsinceR(x)
¬¬
isafinitarystatementthisimpliesR(x).
TheargumentismodernversionofHilbert’sargumentasfound,forinstance,in(Hilbert;
1927,78):
AberauchwersichmitderWiderspruchsfreiheitnichbegnu¨gtundnochweiter-
gehende Gewissensskrupel hat, muß die Bedeutung des Beweises der Wider-
spruchsfreiheit anerkennen, na¨mlich als einer allgemeinen Methode aus Be-
weisen fu¨r allgemeine Sa¨tze vom Charakter etwa des Fermatschen Satzes, die
mitHilfedere -Funktiongefu¨hrtsind,finiteBeweisezugewinnen.6
6Thee operatorwasatechnicalinventionwithinprooftheory,andHilbertsawitasrepresentingahighlyideal
aspectofmathematics.