Archimedes - Descartes-Hilbert -Tarski

23 Pages·2015·0.31 MB·English

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