The Absolute Arithmetic and Geometric Continua
Author(s): Philip Ehrlich
Source: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association,
Vol. 1986, Volume Two: Symposia and Invited Papers, (1986), pp. 237-246
Published by: The University of Chicago Press on behalf of the Philosophy of Science
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The Absolute Arithmetic and Geometric Continual
PhilipEhrlich
Brown University
1. Introduction
With the appearanceof J.H. Conway'sOn Numbersand Games (1976), the mathematical andphilosophicalcommunitieshave much to celebrate. It is Conway'simportant
discoverythatthe familiarDedekindcut andvon Neumannordinalconstructionsarepart
of a more generalconstructionwhich leads to a properclass of numbersembracingthe
reals and the ordinalsas well as many less familiarnumbersincluding -o, co/2,l/co, /o
and co-n,where cois the least infinite ordinal. Conway furthershows that the arithmetic
of the reals may be extendedto the entireclass yielding a real-closed orderedfield (1976,
pp. 40-42); thatis, an orderedfield where every positive element is a square,and every
polynomialof odd degree with coefficients in the field has a solutionin the field. From a
logical standpoint,real-closedorderedfields areprecisely those orderedfields which are
first-orderindistinguishablefrom the orderedfield of reals (Tarski1951), or to put this
anotherway, they are precisely those orderedfields which satisfy the first-ordercontent
of the DedekindCut Axiom as encapsulatedby the algebraicformulationof Tarski's
elementarycontinuityschema. (1951, p. 49).
Conway constructshis real-closedfield by means of the following conventionsand
recursiveclauses.
Construction.2
If L, R are any two sets of numbers,and no memberof L is > any memberof R, then
there is a number (L IR}. All numbersare constructedin this way.
Convention.
If x = {L IR) we write xL for the typical memberof L, and xR for the typical member
of R. For x itself we then write {xL IxR. x = (a,b,c ...I d,e,f...} means that x = {L I
R}, where a,b,c,... are the typical membersof L, and d,e,f... the typical membersof R.
Definition of x > y, x < y.
x > y iff (no xR< y and x <no yL), and x < y iffy > x.
PSA 1986, Volume 2, pp. 237-246
Copyright? 1987 by the Philosophy of Science Association
238
Definition of x = y, x > y, x < y.
x = y iff(x
y and y >x).
x > y iff(x
y and[not] y >x). x < y iff y > x.
Definition of x + y.
x + y = {xL + y, x + yL I xR + y, x + yR.
Definition of -x.
-x = {-xR I -xL}.
Definition of xy.
xy = (xLy + xyL- LyL,xRy + yR xRyRI
I xLy + xyR - xLyR, xRy + xyL - xRyL}. (1976, pp. 4-5).
This particularreal-closedfield, which Conwaycalls No, is remarkablyinclusive,
Indeed,subjectto the proviso thatnumbersbe individuallydefinablein termsof sets of
von Neumann-Bemays-Godelset theorywith Global Choice, henceforthNBG (cf.
Mendelson, 1979, Ch. 4), it may be said to contain 'All NumbersGreatand Small'
(Conway 1976, pp. 3-4) or allfinite, infiniteand infinitesimalnumbers!3The complex
numbersandp-adic numbersare, of course,precludeddue to the ordering. On the other
hand,if one follows the standardpracticeof identifyingthe cardinalN awith the initial
numberoa, a practicethatis warrantedby the Axiom of Choice (cf. Kunen 1980, pp.
27-28), thenNo may be regardedas containingthe cardinals(althoughtheirarithmetic
being defined 'naturally'differsfrom the familiararithmeticof Cantor). Whatis more
importantfrom the standpointof the presentdiscussion, however,is thatin virtueof the
classical Tarskianrelationthatexists betweenreal-closedorderedfields andmodels of
complete(elementary)Euclidean,hyperbolicand ellipticgeometriccontinuaof
dimensionsn > 1 (Tarski 1951, 1959; Szmielew 1959; Schwabhauser1965), the ndimensionalCartesian,Klein and elliptic Klein spaces definedover No (henceforth
Cn(No),Kn(No)and EKn(No))4may be regardedas providing analyticrepresentations,
relativeto theirrespectivegeometries,of the spatialrelationsdefinedover all possible
types of positions near andfar (thatarerepresentablein termsof sets of NBG).
It is importantto emphasizethatthe set-theoreticprovisos affixed to the above
intuitivecharacterizationsof No and its geometriccounterpartsare crucial,since if one
also permitsnumbersthatare individuallydefinablein termsof properclasses, or even
the more multitudinouscollections warrantedin, say, Ackermann'sset theory (cf.
Ackermann1937; Reinhardt1970; and Fraenkelet. at. 1973, pp. 148-153), then even
richerarithmeticand analyticgeometricspectraare definable,albeitnot in NBG. Such
structurescould, for example, be constructedin Ackermann'sset theory, and, of course,
one could always constructmodels of such structuresin termsof sets of ZermeloFraenkelset theory(ZFC) supplementedwith the assumptionof the existence of
sufficiently many inaccessible cardinals;however, these sets, unlike those of NBG, do
not coincide with the sets of ZFC (i.e., the standardsets of the workingmathematician).
Moreover,unlike NBG (or even Ackermann'sset theoryfor thatmatter),these set
theoriescannot be shown to be equiconsistentwith ZFC, a theorywhich, unlike NBG,
only has sets. For referencesas well as an excellent discussion of these mattersthe
readeris referredto (Fraenkelet. al. 1973, pp. 119-153), where NBG = VBG + Global
Choice.
In a series of recent investigations(forthcominga, forthcomingb, and forthcomingc)
the authorestablisheda varietyof resultsthatwere intendedto lend precisionto, and
239
justificationfor, the above intuitivecharacterizationsof No andits Euclidean,hyperbolic,
andelliptic geometriccounterpartsandto clarifythe relationshipthatexists betweenthese
elementarycontinuaon the one hand,andthe classical (i.e. real) arithmeticand geometric
continua (henceforthR, Cn(R),Kn(R)and EKn(R))on the other. Included among these
resultsare novel (categorical)axiomatizationsof the classical continuamakinguse of the
Archimedeanaxiom in conjunctionwith hithertoforeunnoticedcontinuityaxiomsbased
on concepts adaptedfrom the theoryof homogeneousuniversalmodels, a theorylargely
concernedwith the studyof models which intuitivelyhave all possible types of elements.
(Cf. Jonsson's (1956, 1960 and 1965) and Morley-Vaught's(1962).) We believe we are
justified in referringto these axioms as continuityaxioms since in the context of these
axiomatizationsthey areequivalentto any of the more familiarcontinuityconditions
including, for example, those due to Cantor,Dedekindand Hilbert. Workingin NBG
we furthershowed thatif the Archimedeanaxiom is deletedfrom the aforementioned
axiomatizationsone obtains (categorical)axiomatizationsof No, Cn(No),Kn(No)and
EKn(No),respectively. It is the author'scontentionthatthese results may be naturally
interpretedas supportingthe thesis thatwhereasthe classical arithmeticand geometric
continuashouldmerelybe regardedas arithmeticand geometriccontinuamodulothe
Archimedeancondition(that is, modulothefamiliar conditionthatprecludesthe existence
of non-zeronumbers(respectivelynon-degenerateline segments)that are either infiniteor
infinitesimalrelativeto one another),No and its geometriccounterpartsmay be regarded
as absolute arithmeticand geometriccontinua(moduloNBG).
The above thesis, which may be referredto as The ContinuumThesis, rests in parton
the following maximalitythesis implicitin our previousremarks;namely,thatdespite the
ratherpicturesqueflavorof the above intuitivecharacterizations
of No and its geometric
counterparts,these structuresare essentiallywhat the picturesquecharacterizationssay
they are;thatis, in appropriatesenses thatcan be madeprecise, No and its geometric
counterpartsare (moduloNBG) the maximalstructuresof theirarithmeticand respective
geometrickinds,or to put this anotherway, these structuresrespectivelyexhibitall
possible typesof arithmeticand n-dimensionalEuclidean,hyperbolicand elliptic
geometricgradations (moduloNBG). If this were not the case it is hardto see how we
could justify the continuumthesis, for intuitively,an orderedfield F or a Cartesian,Klein
or elliptic Klein space defined over F may be said to have 'no gaps' and therebysatisfy
the fundamentalintuitionunderlyingall theoriesof continuaif, andonly if, they exhibit
'all possible types of gradations.' Of course, as our previous remarkssuggest, in the
context of a formaltheory,any attemptto lend precisionto the idea of 'all possible types
of gradations'is necessarilydoomedto failureand must be replacedby the more modest
attemptto make precise the idea of 'all possible types of gradations(modulosome set of
theoreticalconstraints).' In the classical theory,types of gradationsare theoretically
significantjust in case they areconsistentwith the archimedeancondition,andin the
context of our theorythey aretheoreticallysignificantjust in cast they are (roughly
speaking)definable in termsof sets of NBG. Thus, whereasthe intuitiveimportof the
maximalitythesis is thatthe purportedcontinuahave no holes thatare definablein terms
of sets, the intuitiveimportof the continuumthesis is thatbeing gap-freein this respect
justifies theircharacterizationas continua.
It should be emphasizedthatthe continuumthesis and the maximalitythesis on which
it depends must be regardedas philosophical theses, theses of the same type as Church's
Thesis concerningeffectively calculablefunctions,the Cantor-DedekindThesis regarding
the structureof the classical Euclideanlinearcontinuumor the thesis thatassociatesthe
areaundera curve with the definiteintegral. In each of these cases a proposalis being
put forthto the effect thata particularformalnotionprovidesa satisfactory
characterizationof a vaguely conceived intuitiveidea. This being the case, a rigorous
proof of the identificationis out of the question;at best, one can only marshalevidence to
the effect thatthe formal notioncapturesthe intuitiveidea in questionand, moreover,it is
unlikely thatany non-equivalentformalnotioncould capturethe intuitionto as greatan
240
extent, given the varioustheoreticalconstraintsplacedon it. Ultimately,the acceptability
of any such identificationrests on the clarityand simplicityof the underlyingtheoretical
notions, theirharmonywith the intuitiveideas, and the power andinstructivenessof the
mathematicalandphilosophicalimplicationsderivedin termsof them.
In the pages thatfollow we will providea brief overview of some of the
aforementionedresultsin an attemptto lend initialplausibilityto the maximalityand
of the purportedcontinuathatwill be
continuumtheses. All of the characterizations
discussed are concernedwith theirrespective structuresas models of first-ordertheories.
We hastento emphasize,however, thatin additionto its rich structureas an orderedfield
(or a real-closedorderedfield) No has a distinguishedset-theoreticstructurewhich
emerges from the recursiveclauses in termsof which it is defined, a structurewhose
interpretationwill play a fundamentalrole in the more completeformulationof our theory
to be presentedin the essay referredto in note 1. Thus, in an importantsense, from the
standpointof the presentessay the conceptsof absolutearithmeticand geometriccontinua
andthe conceptsof absolutearithmeticand geometric(elementary)continuamay be used
interchangeably.
With the scope of our essay now having been made clear, we will introducethe
definitionsrequiredfor the formulationof the aforementionedcharacterizationtheorems.
Following theirformulationwe will offer a few remarksin supportof our two theses,
leaving a more detaileddiscussionfor the expandedversion of this essay alludedto
above.
2. DefinitionsandCharacterization
Theorems
A model A for a theoryT in a languageL will be said to be universallyextendingif
for any models B and C of T in L where B is a substructureof A, C is an extension of B,
and the universesof B and C are sets, there is a model C* of T in L thatis a substructure
of A such thatC is isomorphicto C* where the isomorphismis an extension of the
identitymap on B. We will also say thata model A for a theoryT in L is absolutely
homogeneousuniversalif it is absolutelyuniversalwith respectto T, i.e. every model of
T in L can be embeddedin A, and if it is absolutely homogeneouswith respecd to T, i.e.,
given any two substructuresof A thatare models of T in L (whose universes are sets)
and an isomorphismbetween them, the isomorphismcan be extendedto an
automorphismof A, i.e., an isomorphismfrom A onto A.
If T is a first-ordertheorywith an infinitemodel, then the universeof any universally
extendingor absolutelyhomogeneousuniversalmodel of T is necessarilya properclass.
Since truthfor first-orderassertionsis not, in general,definablein class-structuresin
NBG, talk of such models is not, in general,permissible. No improprietywill be
committedin our discussion, however, since all of the first-ordertheorieswith infinite
models what will concernus are VA-theoriesi.e., theories each of whose sentences is Tequivalentto a sentenceof the form Vx1.. .Vxn 1y1... ym0,where 0 is quantifier-free,
and the notion of a class-model for such theoriesis definablein NBG, assumingthatL is
a set. See the author's(forthcominga) for referencesand details.
As the readerwill recall, being orderedfields, R andNo arerelationalstructureseach
consisting of a universe(i.e., a collection of numbers)togetherwith a set of relations
defined on it. Here the sets consist of an orderrelation,the operationsof additionand
multiplicationtreatedas specialrelations,and two distinguishedelements (the additive
andmultiplicativeidentitiesof the fields) likewise treatedas special relations. With this in
mind the relationshipthatexists betweenR andNo can be expressedby means of the
following two theorems.
241
Theorem1: (I) The following sets of axioms constitute(categorical)axiomatizationsof
R; (II) by deleting the Archimedeanaxiom from the following axiomatizationsone
obtains (categorical)axiomatizationsof No.
Axioms for orderedfields (expressed in L = {+, , <, O, 1})
Archimedeanaxiom
Axiom of ArithmeticExtendability
or alternatively
ContinuityAxioms
Axiom of ArithmeticHomogeneousUniversality
wherebythe lattertwo axioms we mean the following:
Axiomof ArithmeticExtendability:The collection of numberstogetherwith the
correspondingarithmeticandorderrelationsdefinedon it constitutesa universally
extendingmodel of the above statedaxioms.5
Axiomof ArithmeticHomogeneousUniversality: The collection of numberstogether
with the correspondingarithmeticand orderrelationsdefinedon it constitutesan
absolutelyhomogeneousuniversalmodel of the above statedaxioms.
Theorem2: Same as Theorem 1, except thatthe axioms for orderedfields are
supplementedwith the additionalaxioms requiredfor the theoryof real-closedordered
fields.
To formulatethe geometriccounterpartsof Theorems1 and 2, we requirethe
following additionalterminology.
For n > 2, let En,Hn and ELnbe the axioms for complete (elementary)n-dimensional
Euclidean,Hyperbolicand elliptic geometryfound in Tarski's(1959), Szmielew's
(1959), and Schwabhauser's(1965), respectively. In these axiomatizationspoints are
primitiveindividuals,andequidistanceand the orderof collinearpoints areprimitive
notions;equidistanceis a 4-place relationand the orderof collinearpoints is a 3-place
relationof betweennessin the Euclideanand hyperboliccases and a 4-place relationof
cyclic order (i.e., separation)in the elliptic case. Also let En(respectivelyHn;
respectivelyEL1)be En (respectivelyHn;respectivelyELn)less the corresponding
(geometric)elementarycontinuityschema, and let Hn beH* supplementedwith the
formulationof Hilbert'saxiom of hyperbolicparallels found in Szmielew's (1961, pp.
134-135). En, Hn and EL~are equivalentto the standardaxiom sets for classical
Euclidean,hyperbolicand elliptic geometryless the correspondingDedekindCut Axiom
and the models of E, Hn and ELnare precisely the models of EF,H' and ELnwhich rule
out the full infinitevarietyof elementaryspatial anomaliesthatareprecludedby the
axioms for classical real geometry. So, for example, in these geometricspaces all angles
can be trisected,all lines joining interiorandexteriorpoints of a circle intersectthe given
circle, and all lines joining points on the opposite sides of a line intersectthe given line.
See, for example, Tarski's(1959, p. 26) for the Euclideancase.
Theorem3: Let n 2 2. (I) The following sets of axioms constitute (categorical)
axiomatizationsof Cn(R),Kn(R)and EKn(R),respectively; (II) by deleting the
Archimedeanaxiom from the following axiomatizationsone obtains(categorical)
axiomatizations of Cn(No),Kn(No) and EKn(No).
242
En (respectivelyHn;respectivelyEL*)
ArchimedeanAxiom (Geometric)
Axiom of GeometricExtendability
or alternatively
J
ContinuityAxioms
Axiom of GeometricHomogeneousUniversality
wherebythe lattertwo axioms we mean the axioms thatdifferfrom theirarithmetic
counterpartsonly insofaras the referencesto numbersand the arithmeticand order
relationsdefined on them arereplacedby referencesto points (i.e., appropriaten-tuples
of numbers)and the correspondinggeometricrelationsdefined on them (i.e., the analytic
interpretationsof the respectiveaforementionedprimitivenotions). See the reference
cited in note 4 for details.
Theorem4: Same as Theorem3, except thatEn,Hn and ELnare employed in place of
En, H' and ELJ,respectively.
3. Mathematico-Philosophical
Remarks
In virtueof Theorem 1, the basis of our confidencein the arithmeticportionof the
maximalitythesis shouldnow be clear;for not only is No an absolutelyuniversalordered
field, but with the sole exception of No (and of course isomorphiccopies of No) given an
orderedfield F therealways exists an orderedfield B (whose universeis a set) which is a
properextension of some orderedsubfieldA of F for which thereis not alreadya proper
extension of A containedin F thatis isomorphicwith B, where the isomorphismleaves
the elements of A fixed. Thus, unlikeNo one can always supplementF with types of
arithmeticgradationsthatit does not alreadyexhibit. By appealingto Theorem3, the
geometricportionof the maximalitythesis can be arguedfor along similarlines, and by
appealingto Theorems2 and 4 respectively,supportfor regardingNo and its geometric
counterpartsas uniquemaximalarithmeticandgeometricelementarycontinua(modulo
NBG) can likewise be obtained.
It is the 'fact' thatNo andits geometriccounterpartsmay be regardedas unique
maximalelementarycontinuamodulothe set-theoreticproviso which providesmuch of
the motivationfor regardingthem as absolutecontinua;being elementarycontinuathey
have a plethoraof the intuitivepropertiesgenerallyattributedto continua,andbeing the
uniquemaximalsuch structuresmodulo whatis at once the most liberaland fundamental
of all theoreticalconstraintsof standardcontemporarymathematicslends a strongair of
legitimacyto theircharacterizationas absolute. In additionto these considerationswe
believe thatthe remarkablestructurealsimilaritythatexists betweenthese structureson
the one handand the classical continuaon the otherlends additionalsupportfor the
continuumthesis. One'sconfidence in the thesis would undoubtedlybe enhancedstill
furtherif it were possible to find a simple axiom, generalizingone of our most basic
intuitionsregardingcontinua,which would single out the purportedabsolutecontinua'
from the wealth of otherelementarycontinua. As the authorshowed in his (forthcoming
b), and as we will now explain, this is indeed possible.
of the purportedabsolutearithmeticcontinuum,one
To obtainsuch a characterization
simply supplementsthe axioms for real-closedorderedfields with the following assertion
(where X and Y are understoodto range over sets).
243
(*) VXVY
(X < Y -- zz (X < (z) < Y)).
Similarly,to obtain a categoricalaxiomatizationof Cn(No)(respectivelyKn(No))one
merely has to supplementEn (respectivelyHn)with the following geometricanalogueof
(*).
(**) VXVY
({aVxVy
(x e X & y e Y -
Baxy & x
?
y) ->
9bVxVy (x e X &y e Y -> Bxby & b ? x & b y)},
where Bxyz is read y lies between x and z (the case when y coincides with x or z not
being excluded). Finally, to obtaina (categorical)axiomatizationof EKn(No)one simply
has to supplementELnwith the elliptic form of (**). See the author's(forthcomingc)
for details.
(*) asserts thatthe collection of numbersis absolutelydense and, intuitively,(**)
assertsthateach line in Cn(No) andKn(No)is absolutelydivisible in the sense thatgiven
any two subsets of points X and Y at which the line can be divided (i.e., intersected),if
X < Y thereis a point lying between X andY at which the line can be divided further.
The elliptic form of (**) has an analogousinterpretation.These axioms may thereforebe
regardedas axioms of absolute densityand divisibilityrespectively. Thus, in virtue of
our earlierremarks,we see thatthepurportedabsolutearithmeticcontinuumis simplythe
uniqueabsolutelydense orderedfield whichpossesses no elementaryalgebraic
limitations,and (exceptfor isomorphicstructures)thepurportedabsoluteEuclidean,
hyperbolicand ellipticgeometriccontinuaare simplythose Cartesian,Klein and elliptic
Klein spaces which admitno elementaryspatial anomaliesand whose lines are absolutely
divisible.
The term 'arithmeticcontinuum'is used to denotethe system of real numbersbecauseit
is held thatthis numbersystem is completelyadequatefor the analyticalrepresentationof
what is known as continuousmagnitude. While we believe it would be rash to asserta
priori thatphysical science will neverhave reasonto rejectthis thesis, therecertainly
seems to be no reason at the presenttime to do so. The theoryof absolutecontinua
sketchedabove is, therefore,not intendedto challengethe role played by the classical
theoryin physical theorizing. Nor do we wish to suggest that,unlike the classical
theory,it provides 'the correct'analysis of continuity. Rather,we merely wish to call
attentionto the fact thatNo and its geometriccounterpartsadmitnaturalinterpretationsas
absolutecontinuaand, moreover,thatthese interpretationslend insight into the structure
of the classical continuaqua restrictedcontinua.
Notes
lThis essay is an expansionof a portionof the materialpresentedby the authorat the
meetings in Pittsburgh.Many of the resultsreportedat thattime have been deleted and
will be discussed in an expandedversion of the originalpaperbeing preparedfor a
forthcomingissue of Syntheseconcernedwith theoriesof continua. Proofs of a few of
the deleted results can be found in Alling and Ehrlich's(1986a and 1986b).
2Forthe purposeof this essay, we are ignoring certainaspects of Conway's
constructionthat would be requiredfor a formaldevelopmentin NBG. See Conway's
(1976, pp. 15-16, 65) for details.
244
numbersgreat and small are equated
3Strictlyspeaking,in these characterizations
with numberswhich are elements of (commutative)orderedfields. This, of course,
includes all the numbersthatare containedin orderedabeliangroupsas well as any of the
otherorderedalgebraicsystems which can be embeddedin an orderedfield.
4Forthe definitions of Cn(F),Kn(F)and EKn(F)where F is an orderedfield see, for
example, (Schwabhauseret. al. 1983, pp. 16, 208 and 213).
5Thecharacterizationof No as a universallyextendingorderedfield was first
establishedby Conway (1976, pp. 42-43). An alternativeproof was providedin the
author's(forthcomingb).
245
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