Rend. Mat. Acc. Lincei s. 9, v. 16143-157 2005) Matematica.(8)

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.