Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica.(4)

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).