Axiomatizing changing conceptions of the geometric continuuum II: Archimedes - Descartes-Hilbert -Tarski JohnT.Baldwin DepartmentofMathematics,StatisticsandComputerScience UniversityofIllinoisatChicago∗ January5,2015 InPartI[Bal14a],wedefinedthenotionofamodestcompletedescriptiveax- iomatizationandshowedthatHP5andEGaresuchaxiomatizationsofEuclid’spolyg- onal geometry and Euclidean circle geometry1. In this paper we argue: 1) Tarski’s axiom set E2 is a modest complete descriptive axiomatization of Cartesian geometry (Section 2; 2) the theories EG and E2 are modest complete descriptive ax- π,C,A π,C,A iomatizationsofEuclideancirclegeometryandCartesiangeometry,respectivelywhen extended by formulas computing the area and circumference of a circle (Section 3); and 3) that Hilbert’s system in the Grundlagen is an immodest axiomatization of any of these geometries. As part of the last claim (Section 4), we analyze the role of the ArchimdedeanpostulateintheGrundlagen,tracetheintricaterelationshipbetweenal- ternativeformulationsof‘Dedekindcompleteness’,andexhibitmanyothercategorical axiomatizationsofrelatedgeometries. 1 Background and Definitions In[Bal14a],weexpoundedthefollowinghistoricaldescription. Remark1.1(Background). Euclidfoundshistheoryofarea(ofcirclesandpolygons) onEudoxus’theoryofproportionandthus(implicitly)ontheaxiomofArchimedes. Hilbertshowsany‘Hilbertplane’interpretsafieldandrecoversEuclid’sthe- oryofpolygonsinafirstordertheory. ∗ResearchpartiallysupportedbySimonstravelgrantG5402. 1TheseaxiomsaredescribedinNotation1.3;thesedatasetsaredescribedasspecificsetsofpropositions fromtheElementsin[Bal14a]. 1 The Greeks and Descartes dealt only with geometric objects. The Greeks regardedmultiplicationasanoperationfromlinesegmentstoplanefigures. Descartes interpreteditasanoperationfromlinesegmentstolinesegments. Onlyinthelate19th century, is multiplication regarded as an operation on points (that is ‘numbers’ in the coordinatizingfield). Webuiltin[Bal14a]onDetlefsen’snotionofcompletedescriptiveaxiomati- zationanddefinedamodestcompletedescriptiveaxiomatizationofadatasetΣtobe collectionofsentencesthatimplyallthesentencesinΣand‘nottoomanymore’. A datasetisacollectionofpropositionsaboutamathematicaltopicthatareacceptedata givenpointintime. Ofcourse,therewillbefurtherresultsprovedaboutthistopic. But ifthissetofaxiomsintroducesessentiallynewconceptstotheareaor,evenworse,con- tradicts the understanding of the original era, we deem the axiomatization immodest. Weillustratethesedefinitionsbyspecificaxiomatizationsofvariousareasofgeometry thatwenowdescribe. Weformulateoursysteminatwo-sortedvocabularyτ chosentomaketheEu- clideanaxioms(eitherasinEuclidorHilbert)easilytranslatableintofirstorderlogic. Thisvocabularyincludesunarypredicatesforpointsandlines, abinaryincidencere- lation, a ternary betweenness relation, a quaternary relation for line congruence and a 6-ary relation for angle congruence. We need one additional first order postulate2 beyondthosein[Hil71]. Postulate1.2. CircleIntersectionPostulate IffrompointsAandB,circleswithra- dius AC and BD are drawn such that one circle contains points both in the interior and in the exterior of the other, then they intersect in two points, on oppositesidesofAB. Notation1.3. WefollowHartshorne[Har00]inthefollowingnomenclature. AHilbertplaneisanymodelofHilbert’sincidence,betweenness3,andcon- gruenceaxioms4. WedenotethisaxiomasHPandwriteHP5fortheseaxiomsplusthe parallelpostulate. By the axioms for Euclidean geometry we mean HP5 and in addition the circle-circleintersectionpostulate1.2. Wewillabbreviatethisaxiomset5 asEG. By 2Mooresuggestsin[Moo88]thatHilbertmayhaveaddedthecompletenessaxiomtothesecondedition specificallybecauseSommerinhisreviewofthefirsteditionpointedoutitdidnotprovetheline-circle intersectionprinciple. 3TheseincludePasch’saxiom(B4of[Har00])asweaxiomatizeplanegeometry. Hartshorne’sversion ofPaschisthatanylineintersectingonesideoftrianglemustintersectoneoftheothertwo. 4TheseaxiomsareequivalenttothecommonnotionsofEuclidandPostulatesI-Vaugmentedbyone trianglecongruencepostulate,usuallytakenasSASsincethe‘proof’ofSASiswhereEuclidmakesillegiti- mateuseofthesuperpositionprinciple. 5Inthevocabularyhere,thereisanaturaltranslationofEuclid’saxiomsintofirstorderstatements. The constructionaxiomshavetobeviewedas‘forall–thereexist’sentences. TheaxiomofArchimedesas discussedbelowisofcoursenotfirstorder. WewriteEuclid’saxiomsforthoseintheoriginal[Euc56]vrs (firstorder)axiomsforEuclideangeometry,EG.NotethatEGisequivalentto(i.e.hasthesamemodels)as thesystemlaidoutinAvigadetal[ADM09],namely,planesoverfieldswhereeverypositiveelementasa squareroot).Thelattersystembuildstheuseofdiagramsintotheproofrules. 2 definition,aEuclideanplaneisamodelofEG:Euclideangeometry. WewriteE2 forTarski’sgeometricalaxiomatizationoftheplaneoverareal closedfield(RCF)(Theorem2.1). 2 From Descartes to Tarski ItisnotourintenttogiveadetailedaccountofDescartes’impactongeometry.Wewant tobringoutthechangesfromtheEuclideantotheCartesiandataset.Forourpurposes, themostimportantistoexplicitly(onpage1of[Des54])definethemultiplicationof line segments to give a line segment which breaks with Greek tradition6. And later on the same page to announce constructions for the extraction of nth roots for all n. The second of these cannot be done in EG, since it is satisfied in the field which has solutionsforallquadraticequationsbutnotthoseofodddegree7. MarcoPanza[Pan11]formulatesintermsofontologyakeyobservation, ThefirstpointconcernswhatImeanby‘Euclid’sgeometry’. Thisisthe theory expounded in the first six books of the Elements and in the Data. Tobemoreprecise,Icallit‘Euclidsplanegeometry’,orEPG,forshort.It isnotaformaltheoryinthemodernsense,and,afortiori,itisnot,then,a deductiveclosureofasetofaxioms.Hence,itisnotaclosedsystem,inthe modernlogicalsenseofthisterm. Still,itisno8 moreasimplecollection of results, nor a mere general insight. It is rather a well-framed system, endowedwithacodifiedlanguage,somebasicassumptions,andrelatively precise deductive rules. And this system is also closed, in another sense ([Jul64] 311-312), since it has sharp-cut limits fixed by its language, its basicassumptions,anditsdeductiverules. Inwhatfollows,especiallyin section1,Ishallbetteraccountforsomeoftheselimits,namelyforthose relativetoitsontology. Morespecifically,Ishalldescribethisontologyas beingcomposedofobjectsavailablewithinthissystem,ratherthanobjects whicharerequiredorpurportedtoexistbyforceoftheassumptionsthat thissystemisbasedonandoftheresultsprovedwithinit.ThismakesEPG radically different from modern mathematical theories (both formal and informal). OneofmyclaimsisthatDescartesgeometrypartiallyreflects thisfeatureofEPG. Inourcontextweinterpret‘composedofobjectsavailablewithinthissystem’ model theoretically as the existence of certain starting points and the closure of each modelofthesystemunderadmittedconstructions. 6HisproofisstillbasedonEudoxus. 7Seesection12of[Har00]. 8Thereappearstobeatypo.Probably‘morea”shouldbedeleted.jb 3 WetakePanza’s‘open’systemtorefertoDescartes’‘linkedconstructions’9 considered by Descartes which greatly extend the ruler and compass licensed in EG. Descartesendorsessuch‘mechanical’constructionsastheduplicationofthecubicas geometric. AccordingtoMolland(page38of[Mol76])”Descartesheldthepossibility ofrepresentingacurvebyanequation(specificationbyproperty)”tobeequivalentto its ”being constructible in terms of the determinate motion criterion (specification by genesis)”. ButasCrippapointsout(page153of[Cri14a])Descartesdidnotprovethis equivalence; thereissomecontroversyastowhetherthe1876workofKempesolves the precise problem. Descartes rejects as non-geometric any method for quadrature of the circle. See page 48 of [Des54] for his classification of problems by degree. UnlikeEuclid,Descartesdoesnotdevelophistheoryaxiomatically. Butanadvantage of the ‘descriptive axiomatization’ rubric is that we can take as data the theorems of Descartes’geometry. For our purpose we take the common identification of Cartesian geometry with”real”algebraicgeometry: thestudyofpolynomialequalitiesandinequalitiesin thetheoryofrealclosedfields. TojustifythisgeometryweadaptTarski’s‘elementary geometry’. ThismovemakesasignificantconceptualstepawayfromDescarteswhose constructionswereonsegmentsandwhodidnotregardalineasasetofpointswhile Tarski’s axiom are given entirely formally in a one sorted language of relations on points. Inourmodernunderstandingofanaxiomsetthetranslationisroutine. From Tarski[Tar59]weget: Theorem2.1. Tarski[Tar59]givesatheoryequivalenttothefollowingsystemofax- iomsE2. Itisfirstordercompleteforthevocabularyτ. 1. Euclideangeometry 2. Eitherofthefollowingtwosetsofaxiomswhichareequivalentoveri) (a) Aninfinitesetofaxiomsdeclaringthateverypolynomialofodd-degreehas aroot. (b) Theaxiomschemaofcontinuitydescribedjustbelow. TheconnectionwithDedekind’sapproachisseenbyTarski’sactualformula- tionasin[GT99]; thefirstordercompletenessofthetheoryisimposedbyanAxiom SchemaofContinuity-adefinableversionofDedekindcuts: (∃a)(∀x)(∀y)[α(x)∧β(y)→B(axy)]→(∃b)(∀x)(∀y)[α(x)∧β(y)→B(xby)], whereB(x,y,z)isthepredicaterepresentingyisbetweenxandz,α,βarefirst-order formulas, the first of which does not contain any free occurrences of a,b,y nor the secondanyfreeoccurrencesofa,b,x. Thisschemaallowsthesolutionofodddegree polynomials. 9ThetypesofconstructionsallowedareanalyzedindetailinSection1.2of[Pan11]andthedistinctions withtheCartesianviewinSection3.Seealso[Bos01]. 4 Remark2.2(Go¨delcompleteness). InDetlefsen’sterminologywehavefoundaGo¨del completeaxiomatizationof(inourterminologyCartesian)planegeometry. Thisguar- antees that if we keep the vocabulary and continue to accept the same data set no axiomatizationcanaccountformoreofthedata. Therearecertainlyopenproblemsin planegeometry[KW91].Buthowevertheyaresolvedtheproofwillbeformalizablein E2. Ofcourse,moreperspicuousaxiomatizationsmaybefound. Oronemaydiscover theentiresubjectisbetterviewedasanexampleinamoregeneralcontext. Inthecaseathand,however,therearemorespecificreasonsforacceptingthe geometryoverrealclosedfieldsas‘thebest’descriptiveaxiomatization. Itistheonly onewhichisdecidableand‘constructivelyjustifiable’(SeeTheorem3.3.). Remark2.3(UndecidabilityandConsistency). Ziegler[Zie82]hasshownthatevery nontrivialfinitelyaxiomatizedsubtheoryofRCF10isnotdecidable. Thus both to more closely approximate the Dedekind continuum and to ob- tain decidability we restrict to planes over RCF and thus to Tarski’s E2 [GT99]. Of course, another crucial contribution of Descartes is coordinate geometry. Tarski pro- vides a converse; his interpretation of the plane into the coordinatizing line [Tar51] underliesoursmudgingofthestudyofthe‘geometrycontinuum’withaxiomatizations of‘geometry’. Thebiinterpretability11 betweenRCFandthetheoryofallplanesover realclosedfieldsyieldsthedecidabilityofE2. Thecrucialfactthatmakesdecidability possibleisthatthenaturalnumbersarenotfirstorderdefinableintherealfield. The geometry can represent multiplication as repeated addition in the sense of a module Z overa butnotwiththefullringstructure. 3 Archimedes: π, circumference and area of circles The geometry over a Euclidean field (every positive number has a square root) may have no straight line segment of length π, since the model containing only the con- structiblerealnumbersdoesnotcontainπ. WeextendbyaddingπeachgeometryEG and E2 and write T when discussing results that apply to either. We want to find a theorywhichprovesthecircumferenceandareaformulasforcircles. Ourapproachis toextendthetheoryEGsoastoguaranteethatthereisapointineverymodelwhich behaves as π does. In this section we will show that in this extended theory there is amappingassigningastraightlinesegmenttothecircumferenceofeachcircle. This goaldefinitelydivergesfroma‘Greek’datasetandisorthogonaltotheaxiomatization of Cartesian geometry in Section 2. Given that the entire project is modern, we give theargumentsentirelyinmodernstyle. 10RCFabbreviates‘realclosedfield’;thesearetheorderedfieldssuchthateverypositiveelementhasa squarerootandeveryodddegreepolynomialhasatleastoneroot. Thetheoryiscompleteandrecursively C axiomatizedsodecidable. Bynontrivialsubtheory,Imeanonesatisfiedbyoneof ,(cid:60),orap-adicfield Qp.ForthecontextofZiegler’sresultandTarski’squantifiereliminationincomputersciencesee[Mak13]. 11Tarskiprovestheequivalenceofgeometriesoverrealclosedfieldswithhisaxiomsetin[Tar59]. He callsthetheoryelementarygeometry,E2. 5 Euclid differs from the modern approach in that the sequences constructed in the study of magnitudes in the Elements are of geometric objects, not (even real) numbers. (Implicitly)usingArchimedesaxiom,Euclidproves(XII.2)thattheareaof acircleisproportionaltothesquareofthediameter. Inamodernaccount,aswesaw in[Bal14a],whenwehaveinterpretedsegmentsOaasrepresentingthenumbera,we must identify the proportionality constant and verify that it represents a point in any model of the theory12. This shift in interpretation drives the rest of this section. We searchfirstforthesolutionofaspecificproblem: isπintheunderlyingfield? WenamedthissectionafterArchimedesfortworeasons. Heisthefirst(Mea- surementofaCirclein[Arc97])toprovethecircumferenceofacircleisproportional to the diameter and begin the approximation of the proportionality constant (which wasn’t named for another 2000 years). Secondly, his axiom is used not only in his work but implicitly13 by Euclid in proving the area of a circle is proportional to the squareofthediameter. Bybeginningtocalculateapproximationsofπ,Archimedesis moving towards the treatment of π as a number. The validation in the theory E2 π,C,A belowoftheformulasA=πr2andC =πdareansweringthequestionsofHilbertand DedekindnotquestionsofEuclidorevenArchimedes. ButthetheoryEG iscloser π totheGreekoriginsthantoHilbert’ssecondorderaxioms. Dicta 3.1. Constants 2 Recall (Section 4.4) of [Bal14a]) that closing a plane under rulerandcompassconstructionscorrespondstoclosingthecoordinatizingorderedfield undersquarerootsofpositivenumbers. Havingnamedanarbitrarypointsas0,1,each elementofthemaximalrealquadraticextensionoftherationalfield(thesurdfieldF ) √ s isdenotedbyatermt(x,0,1)builtfromthefieldoperationsand . InDefinition3.2wenameafurthersingleconstantπ. Buttheeffectismuch differentthannaming0and1). Thenewaxiomsspecifytheplaceofπintheordering ofthedefinablepointsofthemodel. Sothedatasetisseriouslyextended. Now that we have established that each model of T has the surd field F s embeddableinthefielddefinableinanylineofthemodel,wecaninterprettheGreek theoryofproportionalityintermsofcuts.Eachpairofproportionalpairsofmagnitudes determinesacut(E.g.page33-34of[Coo63]).Thenon-firstorderpostulatesofHilbert playcomplementaryroles.TheArchimedeanaxiomisminimizing;eachcutisrealized byatmostonepointsoeachmodelhascardinalityatmost2ℵ0.TheVeronesepostulate (See Footnote 20.) or Dedekind’s postulate is maximizing; each cut is realized, In Hilbert’sversion,thesetofrealizationscouldhavearbitrarycardinality. Definition3.2(Axiomsforπ). 1. Addtothevocabularyanewconstantsymbolπ. Leti (c )betheperimeterofaregular3∗2n-goninscribed14 (circumscribed) n n 12Forthisreason,ArchimedesneedsonlyhispostulatewhileHilbertalsoneedsDedekind’spostulateto provethecircumferenceformula. 13Weadoptthisrathercommoninterpretation. ButBorzacchini[Bor06]suggeststhatArchimedesreal innovationistorequireanadditionalhypothesistocomparestraightsegmentswithcurvedones. 14IthankCraigSmorynskiforpointingoutthatisnotsoobviousthatthattheperimeterofaninscribed n-gonismonotonicinnandremindingmethatArchimedesstartedwithahexagonanddoubledthenumber ofsidesateachstep. 6 inacircleofradius1. LetΣ(π)bethecollectionofsentences(i.e. type15) i <2π <c n n forn<ω. 2. WeextendbothEGandE2. (a) EG denotesdeductiveclosureinthevocabularyτ augmentedbyconstant π symbols0,1,πofEGandΣ(π). (b) E2 isformedbyaddingΣ(π)toE2andtakingthedeductiveclosure. π For the second proposition in the next theorem we use some modern model theory. A first order theory T for a vocabulary including a binary relation < is o- minimalifevery1-aryformulaisequivalentinT toaBooleancombinationofequali- tiesandinequalities[dD99]. Anachronistically,theo-minimalityoftherealsisamain conclusionofTarskiin[Tar31]. Thetwotheorieswehaveconstructedhavequitedifferentproperties. Theorem3.3. 1. EG is a consistent but incomplete theory. It is not finitely ax- π iomatizable. 2. E2 andE2 arecompletedecidableo-minimaltheories. Itisprovablyconsistent π inprimitiverecursivearithmetic(PRA). Proof. 1) A model of EG is given by closing F ∪{π} ⊆ (cid:60) under ruler π s andcompassconstructions. Toseeitisnotfinitelyaxiomatizable,foranyfinitesubset Σ of Σ choose a real algebraic number p satisfying Σ ; close F ∪{p} ⊆ (cid:60) under 0 0 s constructibilitytogetamodelofEGwhichisnotamodelofEG . π 2)Wehaveestablishedthattherearewell-definedfieldoperationsontheline through 01. By Tarski, the theory of this real closed field is complete. The field is bi-interpretablewiththeplane[Tar51]sothetheoryofthegeometryT iscompleteas well.FurtherbyTarski,thefieldiso-minimal.Thetypeovertheemptysetofanypoint onthelineisdeterminedbyitspositioninthelinearorderingofthesurdsubfieldF . s Eachi ,c isanelementofthefieldF . Thispositioninthelinearorderof2π inthe n n 0 linearorderonthelinethrough01isgivenbyΣ. ThusT ∪Σiscomplete. TheproofthatconsistencyisprovableinPRA([Bal14b])usesdifferentmeth- ods. Theargumentsof([Sim09]IX.3.18)showthetheoryofrealclosedfieldsisprov- ablyconsistentinWKL ;SincethecompactnesstheoremisalsoprovableinWKL 0 0 thisallowsustoextendtotheconsistencyofE2. AndtheconsistencypassestoPRA π sinceWKL isconservativeoverPRAforπ0-sentences. 0 1 3.3 15LetA ⊂ M |= T. AtypeoverAisasetofformulasφ(x,a)wherex,(a)isafinitesequenceof variables(constantsfromA)thatisconsistentwithT. ItisovertheemptysetiftheelementsofAare definablewithoutparametersinT (e.g.theconstructiblenumbers).HerewetakeT asEG. 7 Dicta 3.4 (Definitions or Postulates I). In Definition 2.5 of [Bal14b] we extend the ordering on segments by adding the lengths of ‘bent lines’ and arcs of circles to the domain. Two approaches16 to this step are a) our approach to introduce an explicit but inductive definition or b) add a new predicate to the vocabulary and new axioms specifyingitbehavior. ThesecondalternativereflectsinawaythetropethatHilbert’s axiomsareimplicitdefinitions. Butthischoiceisnotavailablefortheinitialaxioma- tization. Itisonlybecausewehavealreadyestablishedacertainamountofgeometric vocabularythatwecantakechoicea). Cruciallythedefinitionofbentlines(andthus theperimeterofcertainpolygons)isnotasingledefinitionbutaschemaofformulas φ definingthepropertyforeachn. OurdefinitionismorerestrictedthanArchimedes n whoreallyforeseesthenotionofboundedvariation. In the following, T denotes either EG or E2. To argue that π, as implicitly defined by the theory T , serves its geometric purpose, we adda new unary function π symbolC mappingourfixedlinetoitselfand(Definition2.6of[Bal14b])satisfyinga schemeassertingthatforeachn,C(r)isbetween(inthesenseofthelastparagraph) theperimeterofaregularinscribedn-gonandaregularcircumscribedn-gonofacircle withradiusr. AnyfunctionC(r)satisfyingtheseconditionsiscalledacircumference function;wecallC(r)thecircumferenceofacirclewithradiusr. The justification of the area formula is simpler because we do not need the stepofextendingtheorderingonstraightlinestoarcs. FollowingHilbertwenotedin Section4.6of[Bal14a]thatifonepolygonisincludedinanotherithasasmallerarea. So the approximation is less of an issue than for circumference and we just need the calculationinthesurdfieldthatestimatesofπ computedfromthestandpointsofarea andcircumferenceareinthesamecutinF . s Lemma 3.5. Let, for n < ω, I and C denote the area of the regular 3×2n-gon n n inscribedorcircumscribingtheunitcircle. ThenT proveseachofthesentencesI < π n π <C . n WeextendthetheorytoincludeadefinitionofC(r)andA(r). Definition3.6. 1. ThetheoryT istheextensionoftheτ ∪{0,1,π}-theoryT π,C π obtainedbytheexplicitdefinitionC(r)=2πr. 2. ThetheoryT istheextensionoftheτ ∪{0,1,π,C}-theoryT ,obtained π,C,A π,C bytheexplicitdefinitionA(r)=πr2. Asanextensionbyexplicitdefinition,E2 iscompleteando-minimal. Our π,C definitionofT thenmakesthefollowingmetatheoremimmediate. π Theorem3.7. InT2 ,C(r) = 2πr isacircumferencefunction(i.e. satisfiesallthe π,C conditionsι andγ )andinT2 ,A(r)=πr2isanareafunction. n n π,C,A 16Morespecifically,wecoulddefine<ontheextendeddomainorwecouldaddan<∗tothevocabulary andpostulatethat<∗extends<andsatisfiesthepropertiesofthedefinition. 8 The proof for circumference shows that for each r there is an s ∈ S whose length,2πr islessthantheperimetersofallinscribedpolygonsandgreaterthatthose oftheinscribedpolygons. Wecanverifythatbychoosingnlargeenoughwecanmake i andc asclosetogetheraswelike(moreprecisely,forgivenmdifferby<1/m).In n n phrasingthissentenceIfollowHeath’sdescription17 ofArchimedesstatements, ”But hefollowsthecautiousmethodtowhichtheGreeksalwaysadhered;heneversaysthat givencurveorsurfaceisthelimitingformoftheinscribedorcircumscribedfigure;all thatheassertsisthatwecanapproachthecurveorsurfaceasnearlywasweplease”. WehavenotestablishedthearclengthclaimforeacharcinC forevenoner. r WewillaccomplishthattaskinLemma3.10. InanArchimedeanfieldthereisauniqueinterpretationofπandthusaunique choiceforacircumferencefunctionwithrespecttothevocabularywithouttheconstant π. By adding the constant π to the vocabulary we get a formula which satisfies the conditionsineverymodel. Butinanon-Archimedeanmodel,anypointinthemonad of2πrwouldequallywellfitourconditionforbeingthecircumference. Wehaveextendedourdescriptivelycompleteaxiomatizationfromthepolyg- onal geometry of Hilbert’s first order axioms (HP5) to Euclid’s results on circles and beyond. Euclid doesn’t deal with arc length at all and we have assigned straight line segmentstoboththecircumferenceandareaofacircle. Sothiswouldnotqualifyas amodestaxiomatizationofGreekgeometrybutonlyofthemodernunderstandingof theseformulas. Thisdistinctionisnotaproblemforthenotionofdescriptiveaxioma- tization. Thefactsaresentences. Theformulasforcircumferenceandarenotthesame sentencesastheEuclid/Archimedesstatementintermsofproportions. WecouldhoweverbetterarguethatEG ismodestaxiomatizationofEu- π,A clid, than EG given the absence of arclength from Euclid (and the presence of π,C,A VI.1). Now we extend our goals to find a modest descriptive axiomatization of the modernconceptionofanglemeasure. Thiswillilluminatethedelicatedependenceof descriptiveaxiomatizationonthevocabularyinwhichthefactsareexpressed. Dedekind(page37-38)observesthatwhatwewouldnowcalltherealclosed fieldwithdomainthefieldofrealalgebraicnumbersis‘discontinuouseverywhere’but ‘allconstructionsthatoccurinEuclid’selementscan...bejustasaccuratelyeffected asinaperfectlycontinuousspace’. Strictlyspeaking,forconstructionsthisiscorrect. Buttheproportionalityconstantbetweenacircleanditscircumferenceπisabsent,so, evenmore,notbothastraightlinesegmentofthesamelengthasthecircumferenceand thediameterareinthemodelWewanttofindatheorywhichprovesthecircumference andareaformulasforcirclesandcountablemodelsofthegeometryoverRCF,where ‘arclengthbehavesproperly’. Mueller(page236of[Mue06])makesan important pointdistinguishingthe 17Archimedes,MenofScience[Hea11],introductionKindlelocation393. 9 useofcutsinEuclid/EudoxusfromthatinDedekind. One might say that in applications of the method of exhaustion the limit is given and the problem is to determine a certain kind of sequence con- verging to it, ...Since, in the Elements the limit always has a simple de- scription,theconstructionofthesequencecanbedonewithinthebounds of elementary geometry; and the question of constructing a sequence for anygivenarbitrarylimitneverarises. Thisdistinctioncanbeexpressedinanotherway. Wespeakofthemethodof Eudoxus: a technique to solve certain problems, which are specified in each applica- tion. Incontrast,Dedekind’spostulateprovidesasolutionfor2ℵ0 problems. Descartes eschews the idea that there can be a ratio between a straight line segmentandacurve. As[Cri14b]writes,”Descartes18 excludestheexactknowability oftheratiobetweenstraightandcurvilinearsegments”: ... laproportion,quiestentrelesdroitesetlescourbes,nestpasconnue,et mesmeiecroynelepouvantpasestreparleshommes,onnepourroitrien concluredelquifustexactetassur. Wehavesofar, inthespiritofthequotefromMueller, triedtofindthepro- portionality constant only for a specific proportion. In the remainder of the section, we consider several ways of systematizing the solution of families of such problems. First,usingthecompletenessofE2,wecanfindamodelwhereeveryangledetermines an arc that corresponds to the length of a straight line segment. Then we consider several model theoretic schemes to organize such problems. Before carrying out the constructionweprovidesomecontext. Birkhoff[Bir32]introducedthefollowingaxiominhissystem19. POSTULATE III. The half-lines (cid:96),m, through any point O can be put into (1,1) correspondence with the real numbers a(mod2π), so that, if A (cid:54)= O and B (cid:54)= O are points of (cid:96) and m respectively, the difference a −a (mod2π)is∠AOB. m (cid:96) This is a parallel to his ‘ruler postulate’ which assigns each segment a real numberlength. Thus, Birkhofftakestherealnumbersasanunexaminedbackground object. Atoneswoophehasintroducedmultiplication,andassumedtheArchimedean 18Descartes,Oeuvres,vol. 6,p. 412. CrippaalsoquotesAverrosasemphaticallydenyingthepossibility ofsucharatioandnotesVietaheldsimilarviews. 19ThisistheaxiomsystemusedinvirtuallyallU.S.highschoolssincethe1960’s. 10