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

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.