Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica. D On Axiomatic Foundations Comm

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

Rend.Mat.Acc.Linceis.9,v.16:143-157(2005)

Matematica.ÐOnAxiomaticFoundationsCommontoClassicalPhysicsandSpecialRelativity.Nota(*)diALDOBRESSAN.

ABSTRACT.Ð

(i)TheclassoftheaxiomaticfoundationsmentionedinthetitleiscalledAxFound;anditsstructureistreated

intheintroduction.

(ii)ThisconsistsofPartsAtoGfollowedbytheReferences.

(iii)In[17]Bressan'smodallogicistreatedinaconsciouslynon-rigorousway.Insteadhere,aswellasAx

Found,ithasarigoroustreatment.Suchatreatmenthadbeenappreciatedbythemathematicalphysicist

C.Truesdellin[62].

(iv)In1953Truesdellhadaremarkableintuition,whosecorrectnessappearedonlyin1962,fromBressan's

monograph[3].

(v)AsaforeignmemberoftheLinceiAcademy,Truesdellsupportedsomelogicalfeatures,absentinhis

school,andhegaveM.Pitteria``confidentialcopy''involvingthisfact.

(vi)SincethusthepresentrigoroustreatmentofBressan'smodallogicappearsstronglysupportedby

Truesdell,itwasnaturaltodedicatethepresentworktohismemory.

(vii)Intheintroductiononesaystohaveprovedcertainresults(whoseproofdoesnotappearthere)concerning

rationalmechanicsorBressan'smodallogictreatedrigorously.

KEYWORDS:AxiomaticFoundations;SpecialRelativity;ClassicalPhysics;ContinuousMedia.

RIASSUNTO.ÐSuifondamenticomuniallafisicaclassicaeallarelativitaÁristretta.

(i)LaclassedeifondamentiassiomaticimenzionatineltitoloeÁdettabrevementeAxFound;edeÁtrattata

nell'introduzione.

(ii)QuestaconsistenellePartsA,...,GseguitedalleReferences.

(iii)In[17]lalogicamodalediBressaneÁtrattatainmodoconsciamentenonrigoroso.Invecequiessa,alparidi

AxFound,haunatrattazionerigorosa.UnataletrattazioneerastataapprezzatadalfisicomatematicoTruesdellin[62].

(iv)Nel1953Truesdellebbeunanotevoleintuizione,lacuicorrettezzarisultoÁsolonel1962,dallamonografia

[3]diBressan.

(v)ComeSociostranierodeiLincei,Truesdellsostennedegliaspettilogici,assentinellasuascuola.Inoltre

diedeaM.Pitteriuna``confidentialcopy''involgentetalefatto.

(vi)LapresentetrattazionerigorosadellalogicamodalediBressanapparepercioÁfortementesostenutada

Truesdell.Eraquindinaturalededicareallasuamemoriaillavoroinviato.

(vii)Nell'introduzionesidicediaverottenutocertirisultati(lacuidimostrazionenonappareivi)concernentila

MeccanicarazionaleolalogicamodalediBressantrattatarigorosamente.

CONTENTS

PartA:Ontheclassoftheaxiomaticfoundationsmentionedinthetitle.......................PartB:MainreasonsforchangingsomepartsofSection2in(Br&Mont's D)[17].Astrangebutlucky

situation................................................................

PartC:Reasonsforusing``AxFound''andforwritingthepresentworkinmemoryofProf.C.Truesdell

PartD:SomeresultsreadyforinclusioninAxFound.AveryseriousdifficultyovercomebyA.Zanardo144145145150

(*)Pervenutainformadefinitivaall'Accademiail29settembre2005.

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

144A.BRESSANPartE:OnMontanaro'scollaborationtoBr&Mont,substantiallywithinhisdegreethesis;andonhis

subsequentpapers,inpartveryimportantorevensurprising.........................151

PartF:Sometechnicalpreliminaries.Howassertionscanbereferredtoandhowcontentsareusedinthe

Introductionorincontributingworks.Onsymbolsused,e.g.,inspeakinginEnglishaboutworks

writteninItalian.OnAbbreviations............................................151

PARTA:ONTHECLASSOFTHEAXIOMATIC

FOUNDATIONSMENTIONEDINTHETITLE

Thereisauniqueincreasingseriesofinstantst0;t1;t2;...suchthat,forr 0;1;...,attrsomeassertionsbelongingtoaxiomaticfoundationsbegintobeknown.TheseassertionsconstituteaworkwrcalledcontributiontoAxFound.Theworksw0;w1;...arebrieflydenotedby``ContrWs''andforpracticalmotivesseeAS2andAS2;1below.Ofcourse,forr 0;1;2;...,attrthecontributingworkswr 1;wr 2;...,arenotknown.ThestudyofAxFound-see(i)intheAbstract-isstronglybasedonBr&Mont D[7]-see(AS1;3toAS1;7andmainly)AS1;6below-andaimsatimprovingandamplifying[7].AS1Thementionedcontributingworksareconsecutiveinthattheyformauniqueseries(likeiftheywerechaptersorgroupsofchaptersinasamebook.Thisbookisbeingconstructedanditspossibledevelopmentswillgenerallybeatmostimperfectlyknown).

AS1,2Differentcontributionsmightbepublishedondifferentscientificjournals.AS1,3Ishalluse(alsobutnotonly)`Pitteri03'asanabbreviationforPitteri'spaperpublishedontheJournalofElasticity72,241-261,2003.Thisfactalsoappearsfrom[60].

AS1,4Inref.(i.e.reference)[16](onp.261)ofPitteri2003itisshownhowtoobtainanextendedversionofit,whichIshalldenoteby`ExtendPitt'-e.g.,inordertoobtainExtendPittonecanalsosee[61].

AS1,5InsomeentriesoftheReferencesofthepresent§1,someabbreviationsareexplicitlyaddedwithinparentheses:e.g.,suchadditionsareperformedin[3,4]and[5]bymeansof` DMet',` DGIMC',and` DBressan1974'inordertorespectivelyintroducetheabbreviations`Met',`GIMC'and`Bressan1974'.

AS1,6Besidesusing`Br&Mont'for[17],Icanabbreviate[48](onp.27)inExtendPitt,i.e.,(theonly)Truesdell's654pagebook,by`Truesdell1984'.

AS1,7MoredetailsonabbreviationscanbefoundinPartF.

AS1,8Letusexplicitlyremarkthat,e.g.,withinAS2orAS2;3,AShastobereadasassertionorsubassertionrespectively.

FurthermoreImeaneverysuchassertionasanassertionsetoranassertionconjunction;andIregardeverysubassertionASr;sasanassertionheadedbyASr(forinstancebelongingtoASr).

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

ONAXIOMATICFOUNDATIONSCOMMONTOCLASSICALPHYSICS...145

PARTB:MAINREASONSFORCHANGINGSOMEPARTSOF

SECTION2IN(BR&MONT'S D)[17].ASTRANGEBUTLUCKYSITUATION

AS2Ontheonehand,thearticleBr&Mont-seeAS1;6-treatsparticlesystems(withoutconstraints),isoftheMach-PainleveÂtype(1),andisbasedonA.Bressan'smodallogic(althoughforpracticalmotivesthislogicissometimesconsciouslytreatednon-rigorouslyinBr&Mont-seeftn.7onp.168there-).

AS2,1Furthermore,IbelievethattheaforementionedconscioustolerationofmistakesofmodallogicwasevencompulsoryinBr&Mont,inordernottocompel,e.g.,rigorousmathematicalphysicistsinterestedinBr&Mont(asaworkonmechanics)toknowmy(powerfulbutcomplex)theoryofmodallogic(presentedinGIMC-seeAS1;5-).AS3Ontheotherhand,inthepresentworkthelogicalmodalcalculusMCypresentedinGIMCisinsteadcarefullytakenintoaccount(andalsowithsomeusefulresultsformechanicsappreciablebythewellknownmathematicalphysicistC.Truesdell-seeAS7belowinpartC).

AS4However,inAxFound-seeAS1-acontributingworkwillperformsomechangesimprovingBr&Montnotonlyinconnectionwith(modal)logic;andthisholdsespeciallyforDef.2.3andthesubsequentpartofSect.2.FurthermorethesechangesareinaccordwiththeproofswritteninSects.3to8forthetheoremsconsideredinBr&Mont,whilethesetheoremsaswellastheirproofsstronglycontrastwiththeoriginalversionofDef.2.3andthesubsequentpartofSect.2.Thisisthe(strange)situationmentionedin(PartB)'stitle.

AS4,1NeitherTruesdellnorPitteriattendedanytechnicalcourseonmodallogic.Howevertheyusedthislogicintuitivelyandcorrectly.

AS4,2Inparticular,intheyear2004InotedaremarkableintuitionofTruesdellthathewroteascontributor,withina(polemic)ftn.inapaperappearedin1953-seeAS6belowinpartC.

AS4,3InspiteofAS3,byAS4toAS4;2(henceAS6),IthinkthatalsoreadersnotinterestedinmodallogicmaybeinterestedinmanycontributionsofminetoAxFound.

PARTC:REASONSFORUSING``AXFOUND''ANDFORWRITING

THEPRESENTWORKINMEMORYOFPROF.C.TRUESDELL

(1)In(PartB)'stitlesomechangesneededbyBr&Montarementioned,howevertogether

(1)(a)Abookorpaper,e.g.Met D[3]issaidtobeaÁlaMach-PainleveÂoroftheMach-PainleveÂtype,ifinitnotionssuchasmassandforce(andpossiblyalso,e.g.,inertialspaceand(inertial)instant)arenotregardedasprimitives,unlikewhathappensinmostworks.

(b)E.g.MetisaÁlaMach-PainleveÂatahighlevel,especiallywithrespecttoPainleveÂ1922,i.e.[59].

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

146A.BRESSAN

withastrangesituation(explainedinAS4)thatstronglyweakensthesechanges'importance.

(2)HenceBr&Montsubstantiallyis,Ithink,themoreadvancedamongour(published)articlesonparticlesystems.

(3)IthinkthatElimPrF-see[16]intheReferencesfollowingPartF-isoneamongmybestresultsoftheMach-PainleveÂtype(seeftn.1placedonAS2);itwasfoundinSept04anditisyetunpublished.

(4)Thedifferencebetweenthetitleofthepresentwork(AxFound)andBr&Mont'sisnotsubstantial;butonlyinAxFoundbothtermssuchas``kinematics''or``dynamics''failtoappear(essentially).Thus,briefly,(3)isassertedinthat[16],abbreviatedbyElimPrFandconcerningPrF(proprietaÁfisica,i.e.,physicalproperty),allowsustosomehoweffectivelyperformtheeliminationmentionedin[16]itself.

AS5Letmeaddthatthenotionofphysicalnecessity,which(briefly)isapossiblesenseforthelogicalsymbolN( Dp)ofthelanguageMCy(presentedinGIMC)(2)andwhichstronglyaffectsmanyworksofA.Bressanand,e.g.ofhispupils,isimportant(e.g.inMet)withinbothkinematicsanddynamics.ObviouslyitisalsoessentialinElimPrF.AS6In1953,Truesdellcommunicatedanarticle,preciselyreference[24]inExtendPitt,andinasourcefootnoteheexpressedacompletedisagreement(sharedbyG.Hamel)withthisarticle;butTruesdellexplainedthat``publicationofthisarticlemayarousetheinterestofstudentsofmechanicsandlogicalike,thusperhapsleadingtoapropersolutionofthisoutstandingbutneglectedproblem''.Iregardedtheabovephraseitalicizedbyme,asaremarkableintuitionbecausethefirstpapergivingaparticularsolutiontotheaboveproblemismy(long)1962-articleMet;andittotallycomplieswiththatphrase.

AS7BesidesAS6,letmenotethatinhis1986-letter(inItalian)totheAccademiadeiLincei,Truesdellsupportedaresearchprogram(alreadystartedbyme);andhestronglyappreciatedthelogicalfeaturesofsomearticlesofmine(orofmypupils).Hencethepresentwork,whichoften(rigorously)dealswithabovelogicalfeatures,appearsstronglysupportedbyhim-seeAS7;2-.ThereforeIregarditcorrecttowriteitinmemoryofprof.C.Truesdell.

AS7,1(i)TruesdellgavetoPitteria`confidentialcopy'ofhisaforementioned1986-letter(whichiswritteninItalian)(3).Furthermoreitsmainpartwrittenbymebelowissubstantiallyincludedinitsversionprintedinitem(D)onpp.21-22ofExtendPitt.

(2)PhysicalnecessityisarigorousandstronglyspecifiednotionrelatedtoA.W.Burk'snotionofcausalimplication-see[18].

(3)Muchinformationrelatedwiththislettercanbeobtainedinthefollowingway.Consider(1o)thenotionofphysicalnecessity-seetheprecedingfootnote-(2o)theextendedversionofPitteri2003-see[61]-.Inthisseeinparticularftn.3onp.19,thesecondparagraphofSect.5(onp.19),furthermore(onpp.19to21)theitems(A),(B)and(D).

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

ONAXIOMATICFOUNDATIONSCOMMONTOCLASSICALPHYSICS...147

RemarkthatthisstronglyjustifieswhatIhavewrittenhere.Furthermore(ii)theaforementionedversionprintedin(D)belongstothearticlePitteri2003-seeAS1;3-printedinExtendPitt.(iii)LastlyInotethatitsmainpartwrittenbelowgivesusvarioususefulinformation(possiblynotdirectlyrelatedwithTruesdell'sletter)evenifitfailstobecompletedbyExtendPitt,sothatitisnotfullyunderstandable.

AS7,2(i)LetmespecifythattheMemoriamentionedinassertion(a)belowistheMemoriaLinceamentionedin[12]onp.24ofExtendPitt.Inhis1986-letterTruesdellstronglysupportedthismemoriaasastartingpointfortheextensionofMettocontinuousmedia(4).(ii)Truesdell'sappreciationofBressanappearsonp.259oftheworkPitteri2003-seeAS1;3-fromline-14toline-6,aswellasfrom(D)onp.21ofExtendPitt(mainlyfrom(D)'spart(e)).Furthermore(iii)forhisappreciationofBressaningeneral,onecanseeTruesdell1984-seeAS1;7-and,e.g.,item14intheContentsonitsp.503.There`FearofRealWorkinFormalLogic'ismentioned;andthementionedworkisperformedbyBressan.Incidentally,toseeN.Belnap'sobservationin10withinpagexxivofGIMC'sprefacemaybeusefultoo.(iv)LastlyTruesdell'sappreciationofBressanalsoappearsfrom(somepartsof)thepp.533to534inTruesdell1984.(Heknew,e.g.ftn.7onp.19ofExtendPitt).ThesocalledTruesdell'svolumeisthevolumeonwhichPitteri2003ispublished,i.e.thevolumecontainingthefollowing

SymposiumonrecentadvancesandnewdirectionsinMechanics,ContinuumThermodynamics,andKineticTheoryinMemoryofCliffordA.TruesdellIII,withinthe14thU.S.A.NationalCongressofAppliedMechanics,Blacksburg,June23-28,2002.

AS7,3Belowisourmainpart(a)to(h)ofTruesdell's1986-letter:themainpoints(a)to(f)(onwhichftns.(a)to(c)areplacedforexplanations)andTruesdell'sadditions,mainly(g)to(h).

(a)ThesubjectofthisMemoriabelongstoHilbert'ssixthproblem,`MathematischeBehandlungderAxiomederPhysik...',whichpresentlyisonlypartiallysolved.TruesdellmentionstheremarkablecontributionsgiventoitssolutionÏilhavybyNoll,Williams,Gurtin,Appleby,SÂ,Ziemer,Matolecsi,amongothers;thenheadds:(b)Allofthemhaveusedonlytheprocedureofmathematicalanalysis.Theiraxiomsareproposedasnecessaryconditionstoconstructmathematicalstructuresfitforconsistentandclearapplicationstophysicalsystemsandpresent-daytechnology...

(4)Onp.19ofExtendPittseethe1stparagraphofthetextandftn.3(where'[12]'justreferstotheaboveMemoria).

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

148A.BRESSAN

(c)In1962prof.Bressan,consideringonlymass-pointsystems,inadeepworkofwellknowndifficulty,reachedanimpeccablesolutioninthesenseofmathematicallogicandhewasthefirsttodoso.(d)Inhismorerecentstudieshetreatsclassicalmechanicsinauniversethatcontainsmathematicalobjectscalled`observers',and(d')headdsasuitableexistenceaxiomthatwaslacking(5).(e)Inaddition,heconstructsaformalconceptofphysicalpossibilitywhichabstractlyrepresentsaphysicalexperiment(6).(f)Presentlyprof.Bressaniscompletinghislogicalaxiomatizationofmechanicswithsomeofhispupils(Montanaro,Pitteri,etc.),andinawaycapableofincludingthemechanicsofcontinuousmediainthesenseofNollandhisfollowers.(g)AfteradigressionTruesdellrecommendsthepublicationofaBressan'spaper,theonethatappearedin1987as[10],because`itcarriesoutaremarkablestep'intheprogrammentionedin(f),themaincircumstancesrelevantforithavingbeensketchedin(a)to(e).(h)LastlyTruesdelladdsthatBressanistheuniqueperson(asfarasheknows)capableofmasteringmathematicallogic,rationalmechanics,electromagnetismandspecialrelativity;inhisopinionItalyandtheAccademiadeiLinceicanbeproudofhim.

ÃÃÃ

Here,andaswellasinhisbookTruesdell1984,Truesdellmentionsfieldsoutsidephysics,likelogicandphilosophyofscience.HisremarksaboutBressan'shighstandinginthesefieldsgetaveryauthoritativesupportbyearlierassertionsofthephilosopherofscienceN.BelnapinhisForewordtoGIMC,[...](7).

(5)ThelackingaxiomisAmmissione10.2inMet,p.106(alsorecalledinGIMC,footnote3onpp.110-111).Thisisframedasapossibilitycondition,but(d'),whereexistenceaxiomsubstantiallystandsforpossibilityaxiom,iscorrectbecauseofsomepeculiartechnicalfeaturesofMet;andthisshowsthatTruesdellhadgraspedthemaintechnicalsemanticfeaturesofthatpaper.

(6)Thisconstruction,performedinMetandrefinedlaterinGIMCandBressan1985,agreeswiththeviewsofHamel1908andHamel1927.

(7)Inhisforeword,pp.xiiitoxxvofGIMC,BelnappreliminarysaysthatGIMC`isthemostimportantcontributiontodateconcerningtheintroductionofquantifiersintomodallogic.Itsurpassesanyarticleorbookinthegeneralityofitsconceptions,thedegreeoftheirdevelopmentandtheprofundityoftheiranalysis'(p.xiii,line3).BelnapespeciallypointsoutBressan's`newanalysisofpredication'(p.xiv,line10).

Inthesubsequentsections1to8Belnapdetailsandmotivateshispreliminaryassertions.Forinstancehesays(p.xiv,sect.1,line3):`...that[Bressan'smodallanguage]MLyis-uniquelyamongmodallogics-acompletetypetheorywithnoupperlimitonitstypes,isextremelyimportant'.Indeed,asisbroadlymentionedonp.xvii,line-9,bythelackofthislimitGIMCsolvespositivelytheproblemconsideredinCarnap1954-see[19]-,pp.195to196.AmongBressan's`distinctivesemanticfeatures'(pp.xviiitoxxiii)Belnapmentionsattributes,lambdaabstraction,anddefinitedescriptions.Furthermore,inNewDirections(p.xxiv,lines1to6),BelnapespeciallypointsoutBressan'snotionofabsoluteattribute,andsays:`ThearticulationanddeploymentofthisnotionisextremelyimportanttoBressan'senterprise...'.Thisis

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

ONAXIOMATICFOUNDATIONSCOMMONTOCLASSICALPHYSICS...149AS7,4TheworkElimPrF-see[16]-,mentionedaboveAS5,isstronglyrelatedwithMet,inItalian;andincidentallyBr&Mont(writteninEnglish)appears,e.g.,inthecontentsattheoutsetofthepresent§1.Thenitisuseful,first,togivetheinformationonMetinAS4below,andsecond,towrite(inAS8below)anItalian-EnglishtranslationfollowedbycertaincommentsonsomefeaturesthatIbelieveimportant-seeAS8;1-.

AS7,5MylongarticleMetbelongstoclassicalphysicsanddealswithparticlesystems(mainly)withoutconstraints;anditiswritteninanunusualextensionalpartoftheItalianlanguage,butitcandealwith(themodalnotionof)causalpossibility(8).Furthermore,itisaÁlaMach-PainleveÂ-seeftn.1placedonAS2(inpartB)-.Byusingthesefeatures,inMet(certain)rigorousfoundationsofclassicalparticlemechanicsaÁlaMach-PainleveÂhavebeenstated.

AS7,6Truesdellstronglysupportedaresearchprogram(alreadystartedbyA.Bressan)toextendMettocontinuoussystems.Aswasinpartalreadynoted,thiscanbeeasilyseenfrom(i)abovePitteri'sconfidentialcopyinAS7;1,or(ii)fromSection5ofPitteri2003-seeAS1;3-,p.18,orbetter(iii)fromitsextendedversion:seeftn.3onp.19ofExtendPitt(andthepartofSect.5onpp.18to19).

AS8Hereistheaforementioned(Italian-English)translationrelatedtosomesymbolsusedinMet.1)PE(puntoevento,orcronotopo)EP(eventpoint,orspacetime)2)PM(puntomateriale,orparticle)MP(masspoint,orparticle)3)CMP(casomeccanicamentepossibile)MPC(mechanicallypossiblecase)3')CMP-casoMPC-case4)Ist(istante)Inst(instant)5)PEO(M;g)(puntoeventooccupatodalEPO(M;g)(eventpointoccupiedbythepuntomaterialeMnelCMP-casog,masspointMintheMPC-caseg,pensatocomefunzionePEOdiMeg)regardedasafunctionEPOofMandg)6)Preced(relazionediprecedenzatem-Preced(relationoftimeprecedenceporalefrapuntieventi)betweeneventpoints)7)PrF(proprietaÁfisicanoncinematicaePrF(non-kinematicphysicalpropertyconcernenteunpuntomaterialeadunforaparticleataninstantandinanistanteedinunCMP-caso)MPC-case)

AS8,1Letusnowemphasizethattheabovetranslations5)to7)arechosensuitably,confirmed,forinstance,stly,letusnotethatthecitesmanshipassertioninTruesdell1984-seeAS1;7-,p.532,line-12,issupportedbySection16,p.xxiv,line-13inGIMC,whereBelnapsaysthatGIMC`doesnotitselfcontainanaxiomatizationofphysics(Bressanhaswrittenonthiselsewhereinasomewhatdifferentform...)(hereferstoMet,wheremodalitiesarephrasedinaratherunusualbutextensionallanguage).

(8)Causalpossibilityisconsideredinftn.17onp.22ofExtendPitt.(Seealsoitem(E)onthesamepage).

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

150A.BRESSAN

lookingforwardtoasimplifiedversionofMet.Actually,Metcontainsmuchmorecomplexdefinitionsanalogoustoabove1)to7)(9).

PARTD:SOMERESULTSREADYFORINCLUSIONINAXFOUND.AVERYSERIOUSDIFFICULTYOVERCOMEBYA.ZANARDO

AS9Ihave(intuitively)characterized(certain)primitivenotionscommontoclassicalphysicsandspecialrelativity.ThusarelevantlackinBr&Monthasa(yetunpublished)remedy.

AS10InSect.3onp.169ofBr&Mont,theassertioninvolving(3.3)isanon-trivialtheorem,whoseproofisnotevenhintedat.Now,amongotherthings,a(yetunpublished)rigorousversionofitiswritteninItalian(i.e.ElimPrF).

AS11FromthetitleofPartB,(explainedinAS4),itisobviousthattheproofofsometheoremsassertedinSects.3to8ofBr&Mont-seeAS4-needa(rigorous)completiontakingintoaccountthechangesofDefinition2.3consideredinAxFound.Nowinsomesensethiscompletionisready.

AS12Onp.47ofGIMC,itissaidthat``...onemightobjectthatAS12.21[acceptedinthelogicalcalculusMCy]isconfusing[forcertainreasonsconcerningdescriptionsandhintedonp.47ofGIMC,atline2''andthatitwouldbebettertoreplaceitwithAS12:21Ã[whichisincompatiblewithAS12.21].However,AS12.21waspreferredbecause,e.g.,``Itismoreusefulfor...showing''thevalidityofTheorem63.1(amainresultinGIMC).Unfortunately,later,Bressannotedaseriousnon-acceptableconsequenceofAS12.21-see,e.g.,(a)inthepart(c)ofZanardo2004andthepartofitsp.10belowline10-.Furthermore,the(practically)onlywayseenbyBressantoovercomethisdifficultywastoaskA.Zanardo(whoprofessionallywasalogician)whethersomepreviousresultsofhimcouldbeextendedinacertainway.InZanardo2004averysatisfactoryanswercouldbegiven,bywhichthelastdifficultycompletelydisappearedaswellasthoserelated(above)todescriptionsandAS12.21.

(9)(a)Inthelast10or20yearsIelaboratedarelevantsimplificationofMet(alsowithbroaderhypothesesandbriefernotations)butpracticallyasefficientasitsoriginalversioninconnectionwith(possibleevolutionsof)therealworld.Furthermore,

(b)In2004PitterideliveredaconferenceonMetattheDip.ofStructuralEngineeringoftheUniversityofPisa(Officiallaboratoryforexperimentsonbuildingmaterials);andhewassuccessful.Infactinaletterof19/VII/2004PieroVillaggiowrotetometobemuchimpressedbytheideaspresentedinMetandtheircoherence;furthermorehesuggestedmetopublishanEnglishversionofMet(1962)onsomewellknownscientificjournal,regrettingthatMetwaslittleknown.

(g)InmyreplyIagreedthatMetwaslittleknowntorigorousmathematicalphysicists(andforreasonablemotivesalsovalidforBr&Mont-seeAS2andAS2;1-).HoweverIstressedthatMetwasappreciatedbydifferentscientificcommunities.(ObviouslyVillaggioignored,e.g.,Sect.5ofExtendPitt.

(d)Obviously,inmyreplyIconsidered(a).HoweverIhadnotyetknownmyworkElimPrF,mentionedaboveAS5in§1andstronglyrelatedwithMet.

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

ONAXIOMATICFOUNDATIONSCOMMONTOCLASSICALPHYSICS...151

PARTE:ONMONTANARO'SCOLLABORATIONTOBR&MONT,

SUBSTANTIALLYWITHINHISDEGREETHESIS;

ANDONHISSUBSEQUENTPAPERS,INPARTVERYIMPORTANTOREVENSURPRISING

SinceAxFoundisstronglybasedonBr&MonttowhichA.Montanarocollaborated,(especiallytheendof)thetitleofPartErendersitnaturaltoreserveawholepartofourIntroductionforhim.

AS13Whenthesubjectofadegreethesisisverycomplex,liketheoneofBr&Mont,itisnaturaltoreceivemanysuggestions.

AS13,1However,letusnowspecifythatamongMontanaro'spaperswithoutmycollaboration,e.g.,[29]to[30],[33]and[55]are(atleast)important,[51]to[53]areveryimportant,[34]issurprising,and[48]ismostimportant(InotethatmylongmonographMet-seeAS7;5-iscitedinitsreferences),andMontanaro'sresultobtainedin[33]isevensurprising.

AS13,2Morespecifically,Iwanttoemphasizethat,whileBr&Montisaworkofmechanics(ormathematicalphysics)aÁlaMach-PainleveÂ-seefootnote1onAS2-Montanaro's``surprisingresults''werenotevenknownintheirversionsbelongingtotheusualmechanics(notaÁlaMach-PainleveÂ).

PARTF:SOMETECHNICALPRELIMINARIES.HOWASSERTIONSCANBEREFERRED

TOANDHOWCONTENTSAREUSEDINTHEINTRODUCTIONORINCONTRIBUTINGWORKS.ONSYMBOLSUSED,e.g.,INSPEAKINGINENGLISHABOUTWORKSWRITTENINITALIAN.ONABBREVIATIONS

AS14Here,inAxFound,weusethenotationsofGIMC-seeAS1;5and/or[4]afterPartF,andalsoftn.1(a)placedonAS2-added(a)withsetnotationsverycustomaryin(extensional)mathematicsbutpossiblyincludingmodallogic,andwith(b)divisiondots(followingZanardo1981orbetterZanardo2004).(c)FollowingaboveZanardo'spapersherewe(often)weakentheadmissibilityconditionsusedinGIMConeveryuniversefortheextensionaly-sortedlanguageELy(i.e.theextensionalpartofthemodallanguageMLyconstructedinN3ofGIMC)(10).

(10)(a)TheextensionalsemanticsisbasedonysetsD1toDytobecalledindividualdomains-seeinGIMCfromp.18,line15toformula(9)p.21-.Onp.18,line-10Diisassumedtohaveatleasttwoobjects,oneofwhich,ayi,representsthenon-existingobject(i 1;...;y).Furthermore,DyisidentifiedwiththeclassGofelementarypossiblecasesforMLyÀ1-seep.18,line-2andN5p.16,line4toline8.

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

152A.BRESSAN

(d)ThesemanticsforMLyisbasedontheextensionalsemanticsforELy 1,onthet),andonformula(9)onp.21(11).Inextensionalcorrespondentt th,(tP

connectionwiththenewweakestadmissibilityconditionsforELy 1,thissemanticsturnsouttobeextensional(whileinGIMC,line-1,p.18,itoughttobeessentiallymodal);InowregardasadvantageousthepossibilityoftreatingmodalandextensionalsemanticsuniformlyfollowingZanardo1981andZanardo2004(i.e.,[63]and[66])(12).

AS15InthepresentintroductionwehaveusedauniqueseriesAS1,AS2,...;tomarksomerelevantpartsofit.E.g.,ASrhasexactlyoneentrywritteninboldcharacter;inthepartmarkedbyitnotyetknownexplicationsareintroduced.ForobviousmotivesASrhassomeentriesbothafterandbeforeit.Thelattershowreaderswheretheycanfindahelptounderstandyetunclearwritings.ThesameholdsforthepossiblepartsASr;s,(s 1;2;...)ofeveryelementASrofaboveseries.

AS15,1Intheinitialpagesofthisintroductionitscontentsisincluded.Theanaloguecanbedoneinanycontributingwork.(Eachofthesehasaseriesof,e.g.,ASrstartingwithr 1).

AS15,2Anycontributingworkisdividedinsectionsmarkedby(§r)(brieflysections(§r)),and(§r)isdividedinnumbers(1),(2),...;andeachnumber(s)initems(i),(ii),...E.g.thesub-item(ii3)(ifitexists)canbeusedinsteadof(ii)forgreaterprecision.AS15,3E.g.in(§r)``see(s)(ii)''orin(§r)(s)``see(ii)''means:see(§r)(s)(ii).

AS15,4Formoreclarity,onecanbesuperabundantbothinplacing(above)marksforsections,numbers,anditems(inboldcharacter)andinreferringtothem(innormalcharacter).

AS16E.g.,by`Bressan19RS'IabbreviatetheuniqueBressan'spaperappearedintheyear19RS(providedsuchpaperexists).Ifinthatyearanumbery>0ofBressan'spapersappeared,then(beingy< I)Ilabelthemwiththeindexesa1toayandIabbreviatethemby`Bressanai19RS'(i 1;...;y).

(11)(a)TheclassQIofQIs(quasiintensions)oftypetP ty-seeN2atp.40ofGIMC-isdeterminedby 1ht)isdefinedbytherecursivedefinition(a1)QIty Qyth-seeformula(9)onp.21-wheret t,(tP

hOh (h 1);rh (h 1;r)(r 1;...;y);

hh(t1;...;tn)h (t1;...;tn;y 1);hhh(t1;...;tn:t0)h (t1;...;tn:t0):

(b)HerewementiontwocorrectionstobedoneinGIMC:yy(1st)in(b)onp.19:j1POytn3j1POt1;...;jnPOtn.(2nd)definition(8)onp.21must(obviously)bechangedinto h.

(12)Seep.47ofZanardo1981,fromline7toline10.Theyareimproved,withinZanardo2004,byDefinition1.1,Remark(e)andDefinition1.2inpart(A).ThesedefinitionsallowustoembodyextensionallogicintothelogicalcalculusMCyafterhavingweakened,followingZanardo,therequirementcardG!2usedinGIMCintocardG!1.

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

ONAXIOMATICFOUNDATIONSCOMMONTOCLASSICALPHYSICS...153AS16,1Iftheyear19RSisnotearlierthan2000,theninthecasesy 1andy>1IregardBressanRS=Bressan19RSandBressanaiRS=Bressanai19RS(i 1;...;y)respectively.

AS16,2Theaboverules,wheny 1,include[5]andmoregenerally[1,2]andAS1;7too;andwheny>1theyinclude[9]and[10].

AS16,3Notethat[9]and[10]aretheonlycaseswithlabelswhoseabbreviationsareexplicitlymentionedinourreferences.

AS17TheseriesofworkscalledAxFoundincludesthepresentIntroduction,whichisclassifiedbyx1-seebelowtheprecedingContents.

AS17,1Usuallyaparagraphofagivenpageismeantasacertainsetoflines.

AS17,2Istatethatineverycontributingwork(toAxFound),theboldentryofanyparagraphoftheformxt1-seeAS15-mustbeprintedatthelinewherethisparagraphstarts.Furthermore,(a)followingaUniversityofTorino(Turin),thenotboldASr[ASr;s]can(everywhere)beusedtomentiontheboldASr[ASr;s]-cf.AS15.Hence,inordertomentionaparticularoccurrenceofASr[ASr;s],onehastospecifyitbyusing:``ASr''[``ASr;s''].

AS18TheworksinterestingAxFound,e.g.,contributingworks,formabookbeingconstructed-seeAS1;1-.Furthermore

AS18,1generallytheymainlyhavecertainfeatureshintedatinAS1toAS1;2,andmoreclearlyshowninAS2toAS3,AS4;3,AS7;3(especiallyin(g)),inAS7;5,andlastlyin(theremark)AS7;6dealingwithmylongarticleMet.

AS18,2(a)InAS1onespeaks,e.g.,ofabookbeingconstructed,withdevelopmentsimperfectlyknown.(b)Inthepresentpaperthenotationsx1,x2,etc.,aremeant,Ithink,inamannerwiderthantheusualone,andhencedifferentfromthis.(g)By(a)theanaloguesofthepropertiesconsideredforthesequenceAS1,...;AS1;8inx1holdforeverysimilarsequenceAS1,...,AS1;l,...;writteninanypaperlaterthanx1,(e.g.,calledx2.)(d)Assumenowthatx2hasbeenprinted.Then(d1)anarbitrarymarkASlorASl;m,writteninx1determinesthepagesofx2whereithasbeenprinted.Furthermore(d2)thefollowingadvantagesappear:(1o)thismarkgivesinformationmoreprecisethantheonegivenbythepageswherethemarkisprinted;(2o)itmaybeused,e.g.,inthepaperx2(evenifinx2thismarkisnotused)-cf.thepoint(3)aboveAS5inx1-.(d3)NowfixanarbitrarysPZ>0andconsidertheanalogueforpaperxs,ofwhathasbeensaidaboutx1(thepaperxsbeingsupposedtoexist).(d4)Thus,briefly,asequenceofpapersx(2);x(3);...similartox2canbeconsidered.

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

154A.BRESSAN

ACKNOWLEDGEMENTS

ThispaperisdedicatedtothememoryofProf.C.Truesdell.

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______________

Pervenutail16agosto2005,

informadefinitivail29settembre2005.

DipartimentodiMatematicaPuraeApplicata

UniversitaÁdegliStudidiPadova

ViaBelzoni,7-35131PADOVA

ABSTRACT. D ?i) The class of the axiomatic foundations mentioned in the title is called Ax Found; and its structure is treated in the introduction. ?ii) This consists of Parts A to G followed by the References. ?iii) In [17] Bressan's modal logic is treate

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