Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica.(7)

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).