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Completeness of the Real NumbersAuthor(s): Casper GoffmanSource: Mathematics Magazine, Vol. 47, No. 1 (Jan., 1974), pp. 1-8Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2688754 .

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COMPLETENESS OF THE REAL NUMBERSCASPER GOFFMAN

In memory f 0. F. G. Schilling

Completenessf therealnumbersn thesense ofDedekind everynonemptyetwhichhas an upperbound has a least upperbound)meansthat he setof realnum-bers s complete s a totally rdered et.Completeness f the real numbersn thesenseofCantorfor very auchysequenceofrealnumbershere s a real numberto whicht converges)means theset of real numberss complete s a metric pace.The realnumbersre also complete s a totally rderedgroup.Thiskind of com-pletenesswas used byHilbert 7] in hisworkon thefoundationsfgeometry,ndwill be called completenessn thesenseof Archimedes. ur purposeis to discussthese hree ifferentotionsof completenessnd therelationshipetween hem.1. The definitions. setS is totallyordered f there s an orderrelation< inS which atisfies:(a) foreach pair x,yeS, eitherx < y or y < x, and bothx < y and y < xhold if and only f x = y, and(b) ifx,y,zeS and x < y and y < z thenx ? z. If x < y and x : y we writex < y; ifx < y we also writey ? x, and if x < y we also writey > x. A setA c Sis calleda lower egmentn S ifA =A , A #A and if xe A, y < x impliesye A.A totally rdered et S is said to be completein the Dedekindsense) feach lower

segmentn S has a leastupperbound. The real numbers orm complete otallyordered et.A setS together ith mappingd: S x S -+ R is called a metric pace if(a) foreach x,yeS, d(x,y) > 0, and d(x,y) = 0 if and only f x = y,(b) for each x,yeS, d(x,y) = d(y,x), and(c) for achx, y, E S, d(x,y) + d(y, ) > d(x,z).A sequence x,,} in S is a Cauchysequence f foreach E > 0 there s an N suchthatm,n > N implies (xm,x,,) S; {x,} converges o x iffor ach E > 0 there s anN such thatn > N impliesd(x,Xn) < E. A metricpace S is complete in theCantorsense) ffor achCauchy equence xn} inS there s an x E S suchthat x"} convergesto x. The realnumbers,with he metric (x,y) Jx yJ,orm completemetricspace.A setS is called an abelian totallyorderedgroup f(a) S is a totally rdered et,(b) S is an abeliangroupwithoperation+ and identity , and(c) thetwo structuresre compatible n the sensethat fx, yeS, x < y, andz ES thenx + z < y + z.A totally rdered roup s said to be archimedeanf for very airx, yE S, withx,y > 0,eachmember f thepairis lessthansomemultiple ftheothera multipleof x is nx,i.e., x + + x with n summands, or some n).

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2 MATHEMATICSMAGAZINE [Jan.-Feb.Remark 1. An archimedean otally rderedgroup s necessarilybelian. How-ever,nonarchimedean roups need not be abelian (see [8]).In an abeliantotally rderedgroupS, two elements , yE S, With , y > 0, are

said to be relatively rchimedean f each is less than some multiple f the other.Let S and T = S be abeliantotally rdered roups,withS an ordered ubgroupof T, i.e., the orderrelation nd group operation n S are those nducedby T. Wesay that Tis an archimedean xtension f S iffor ach x E T,x > 0, theres a yE Ssuch thatx and y are relativelyrchimedean.Remark 2. The terminologyrchimedean xtension efers o the relationshipbetween and its extension . f S is archimedean hen o is eachof tsarchimedeanextensions.f S is nonarchimedeanhen tmay have archimedeanxtensions hicharealso nonarchimedean.The situationmay be clarified y thefollowing xample:Example 1. Let S be theset ofordered airs x, y) of integers. rder S lexico-graphically,.e., x,y) < (u, v) if x < u or if x = u and y < v.Let T be the setofordered airsof real numbers, lso ordered exicographically.Withthe usual groupoperation x, y) + (u, v) = (x + u, y + v), S and T are easilyseento be abeliantotally rderedgroupswithS an ordered ubgroupof T. Geo-metrically, is thesetofpoints n the plane and S is the setof latticepointswithinteger oordinates.A pointcomes before nother f tsx coordinates smaller: fthex coordinates rethesame,thepointwhose y coordinates smaller omesfirst.(a) S is nonarchimedean.or, 0, 0) < (0, 1) < (1, 0), and n(O, ) = (0, n) < (1,0)for ach n = 1,2,* .(b) T is an archimedean xtension f S. Let x,.y)E T, x,y) > (0, 0). Theneitherx > 0 or x = 0 and y> 0. Ifx > 0 then x, y) and (1, 0) are relativelyrchimedean.If x = 0 and y> 0 then x, y) and (0,1) are relatively rchimedean. ut (1,0) and(0, 1) are in S.We also note thatpictorially is composed ofNo copies of the integersaid

sidebyside na double sequence nd Tis similarlyomposedof c copiesof thereals.An abeliantotally rdered roup s said to be complete nthe enseofArchimedesif t has no proper rchimedean xtension.THEOREM 1. The real number ystem is complete n the ense of Archimedes.Proof.The proof s rather asy.Let H = R be an archimedeanxtension f R.Let x EH, x > 0. We note that each positivereal r is relatively rchimedeanwithx sincethere s a yER, y > 0, which s relativelyrchimnedeanithx and r is re-

latively rchimedeanwithy. In particular, /2" s relativelyrchimedeanwithx.Accordingly,heres a positive ntegersuch hat j - 1)/2" x < j/2" The ntervalsIn = [(j - 1)/2" /2n] form nestedsequence In} of closed intervals n R withlengths onvergingo zero, so there s a unique real numberyeIn, n = 1,2,n.Then, letting jx - yj denote that one of x - y or y - x whch is > 0, we havejx - y < l/n, for each n, so that nIx-y I < 1. On the otherhand, sinceH is

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1974] COMPLETENESS OF THE REAL NUMBERS 3assumed to be an archimedeanxtension f R, J- y > 0 impliesthat there san n for whichnIx - y > 1. This contradictionorcesJx y = 0 so thatx = yand H = R. Thus,R is complete n the sense of Archimedes.

We note someexamples.(a) The real numbersre complete n all three enses.(b) The integersre complete n the Dedekindsenseand in theCantor sense.They re not complete nthesenseofArchimedesince herealsare a properArchi-medeanextension.(c) The rationals re notcompleten anyofthe three enses.2. Defectsnthedefinitions.onsider hetotally rdered roup T of example1.The members f Tare ordered airsofreal numbers nd T is ordered exicograph-

ically.The set A in T defined yA = [(x, y): (x,y) < (O,n) for omen] = [(x,y): x < O]

is a lower egment. uppose u, v) is an upperbound ofA. Thenu > 0. So, (u, v -1)is an upperboundof A. But (u,v - 1) < (u, v) so that A does not have a leastupper bound.Notonly s thisparticular onarchimedeanroupnotcomplete n the Dedekindsensebutwe have thefollowing eneralfact e.g., [1]):THEOREM 2. If S is a nonarchimedeanbelian totallyordered roup, then isnot complete n the Dedekind sense.Proof. There rex,yeS with < x < y and nx< y for ll n = 1,2, The set

A = [z:z u - x. Butz < nx for ome nx. So u - x < nxand u + x < (n + 2)xwhich lacesu + x inA and violates heassumptionhatu is an upperboundofA.Accordingly, does nothavea leastupperbound.The fact hatno nonarchimedeanroupcan be complete n the Dedekindsensemaybe considered o be a defect n the definition.n section3 we shall remedythisdefect y makingan appropriatemodificationf the definitionf Dedekindcompleteness.We firsturnhowever o the notionofcompletenessn theCantorsense.Therewill lso be a defect erewhichwe describen some detail.We use someelementaryfacts bout ordinalnumbers. he ordertype fa totally rdered etS is theequiv-alencesetto whichS belongs, wototally rdered etsbeingequivalent f there sa one-one rderpreservingorrespondenceetween hem.A totally rdered et S iswellorderedfeachnonemptyubsetof S has a firstlement. he order ypeof awellordered et is called an ordinal number.The smallestordinalnumbers rethefinitenes. The firstnfiniterdinalnumber,heorder ype fthesetofpositive

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4 MATHEMATICSMAGAZINE [Jan.-Feb.integersn theirusual order s designated y 00. Its cardinalnumbers No. Thereare manyordinalnumberswhosecardinalnumber s No. For example, heordinalnumberwo+ I of the well ordered et2, 3,.. ; 1 and the ordinalnumberwo+ woof thewell ordered et1,3,5,. ; 2,4, 6, . bothhave cardinalnumberNo.However,wo is thesmallest rdinalnumberwhose cardinalnumber s NO.The setofall ordinalnumberswhose cardinalnumbersre finite r No formswell ordered et. Its ordinal number s co and co is the smallest rdinalnumberwhose cardinalnumber xceedsNo (e.g., [4]).(a) Theordinalnumber o hasthefollowingroperty: oranyfinite et x*, axof ordinal numbers, ach smaller than wo, there s an ordinal numberox ai,i = 1, .., n with a w0.Thatis to say,foranyfiniteet of finite rdinalnumberstheres a finite rdinalnumber reater hanall ofthem.

(b) The ordinal numberw, has the followingproperty:For any countablyinfiniteetoxl,2, .. ofordinal umbersmallerhanwItheres an a > cx,,,=1,2, ...such thatax w1.Now, let S be a wellordered et whose ordinalnumbers co and letG be theabelian totally rderedgroupof all real functionsn S ordered exicographically.Thismeans hat < y fx(a) < y(ac) or hesmallestc orwhich hey iffer.t followsfrom hepropertyfw0 givenn (b) abovethat, iven ny sequencex1 > x2 > ... >xn> ... > OinG,there s an x > O inG suchthatxn> x for ll n = 1,2, - Indeed,for each n, letaC. e thefirst rdinalnumber orwhich n(Oxn) 0.Then xn(Ocn) 0.There s an oc 01 suchthatc> oc,,, = 1,2,* . Let x be defined yx(a) = 1 andx(fl)= 0 foreachfi oc.Then0 < x 0, there s a yE A suchthatx + y0A. We say twolowersegmentsreequivalentfthey ifferya single lement. henevery quivalence lassconsists fone ortwomembers.n the atter ase,A has a leastupperbound n S and we may

either nclude t or exclude t.For convenience, e shall consider he representativewhichdoes not have a largest lement.We define perations or owersegments.fA and B are lowersegments,heirsum A + B is defined yA + B = [usS:u = x + y, x-A, yeB].

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1974] COMPLETENESS OF THE REAL NUMBERS 5It is easyto seethatA + B is a lower egmentnd that fA and B are dedekindeanthen so is A + B.For lower egments e defineA < B ifA c B.The lower egment eterminedy 0 is designated s 0. Then 0 = [x: x < 0].For any ower egment , a lower egment is called n inversefAifA + B = 0.We now come to the cruxofthe matter boutdedekindeanower egments.

THEOREM 3. A lower egment has an inversefand only f t is dedekindean.Proof. a) SupposeA is dedekindean. et B = [x: - x OA]. We firsthow thatB isa dedekindeanower egment. et x EB andy < x. Since x 0A and -y > - xwe have -y A so thatyeB. Next, etxeA. Then - x B so thatB #S. More-over, fx ? A then x EB so thatB : 0. HenceB is a lower egment.n order o

show that B is dedekindean,et x > 0. There is yEA withy + x 0 A. Then - y-x EB and -y = (-y-x) + x B.WenowshowthatA+ B = 0. Letu 0 suchthatforeach yEA,x + yE A. If A + C 0,there reuc A,vE C suchthatu + v= - x. Butu + xE A.So u + v+ x = 0 andu + v+ x = (u + x) + vEA + C. ThismeansA + C #0.Theorem3 yieldsthe fact that thededekindean ower segmentsn S form nabeliantotally rdered roupS* which s an extension fS (for achxE S the owersegmentAx= [y: y < x] is a dedekindeanower segment).We shall call S* thed-completionf S. Anabelian totally rdered roup s saidto be d-completefall ofitsdedekindean ower egments re of theabove formAx.If S and T are totally rdered etswith c T thenS is said to be dense n T ifx, yE T with < y mplies here s z E S with < z < y. We note hat fS isan abeliantotally rdered roup hen is dense n S*. This s clear ince fA andB are dedekind-ean lowersegmentsn S withA c B, A #& , then heres an x ES suchthatx E Band x OA. We also note that f S' is an extension f S, withS dense niS', thenthere s a one-one orrespondenceetween hededekindeanower egmentsn S andthose nS' obtained nthe naturalway. We leavetheeasy, omewhatmessy, etailsto thereader.

Two interestingactsfollow.PROPOSITION 1. The d-completion fS is d-complete.PROPOSITION 2. S is dense n itsd-completion*,and ifT is anyabelian totallyordered roupsuch that S c Tand S is dense n T, thenT is isomorphicwith nordered ubgroupof S*.Indeed, the property f Proposition2 has been takenas the definitionf d-completeness9] and thismaybe themostnatural orm,

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6 MATHEMATICSMAGAZINE [Jan.-Feb.4. CompletenessntheCantor ense. This form f completenessmay be definedin terms f uniformtructures,ests of closed intervals, r transfiniteequences.We shallusetransfiniteequences utnote hatnview fthe quivalence etween he

Cantor and Dedekind methods hereadermay skipthis ectionfhe so wishes.An abelian totally rdered roup is said to be discretefthere s an xE S, x > 0,such thaty > 0 implies y > x.We note that each nondiscrete beliantotally rderedgroupS has a charac-teristicrdinal * = 4*(S) in terms fwhich onvergences defined. his s establishedin thefollowingway.We first btainthefact hat heres a well ordered ransfinitesequencex., (X 4, ofpositive lementsn S such thatcx ,B 4 implies li x. andfor achx > 0 theres an cx 4 suchthatx. < x. We nowconsider he etofall suchordinals and let4*bethe mallest ne.This s the haracteristicrdinal * = 4*(G).It has a propertyiketheone discussed n Section for he pecialordinals oO ndW1.See [2] fordetails.The propertys as follows:For each i < 4* and ordinals < *, cx , there s a 4 < 4* suchthat 0, there s a 4 < (* suchthat < oc, < 4*impliesIX.-x: I < u. A sequencex,, oc d*, s said to be d* convergento x if foreach u E S, u > 0, there s a 4 < d* suchthat < oc 4* implies - j u. S issaid to be C-completefeach 4* Cauchy equence n S is 4* convergent.Let S be a nondiscretebelian totally rdered roup with haracteristicrdinal

-= *(S). We show thatthere s a one-onecorrespondenceetween quivalenceclassesof * Cauchy equences nddedekindeanower egmentsnS, two * Cauchysequencesx,,

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19741 COMPLETENESS OF THE REAL NUMBERS 75. Hahn groups.We digress ow to mention n importantetofabeliantotallyordered roups. hesegroups,ntroducedyH. Hahn[5], aregroups frealfunctionson a totally rdered et. Let S be a totally rdered et and letH be thesetofall real

functionsn S which re zeroexcept n a subsetof S which s wellorderedwithrespect o the orderrelationn S. Ifx and y areinH itfollows hatx + y is inHsincetheseton which + y is notzero will also be well ordered. t follows hatHis an abelian group.We considerH tobe orderedexicographically,.e.,x < y if forthe smallest for whichx(a) =# (a) we have x(oc) y(a). Then H is an abeliantotallyorderedgroup.We are now readyto statethe Hahn representationheorem:Every abeliantotally rdered roup s isomorphic ith subgroup f Hahngroup.Thistheoremwas proved by Hahn in 1907and is modern n conception nd method fproof.It is a pioneering ffortn a type freasoningwhichhas prevailed speciallyn thethirties nd forties. he proofof Hahn was writtenn a leisurelymanner.For asharp,precise, conomical, ltogethermodernproof,the readermay consultthebook, Grundziuge erMengenlehre,writtenyHausdorffn 1914 [6].

6. Relationshipetween -completenessnda-completeness.ecallthat nabeliantotally rdered roup s a-completecompletenthe enseofArchimedes)f thas noproper rchimedeanxtension.We firstrove heTHEOREM. If S is a-coinplete,hen t is d-complete.Proof.Thisfollows rom hefact,whichwe prove, hatS* is an archimedeanextension fS. We may assumeS nondiscrete. et xE S*, x > 0. Then there s ayES*, y> 0, with y< x. For, if y < x, 0 < y,then ither y < x or2(x - y) < x.Now, suppose2y? x. Since S is dense n S* theres z E S such thatx - y < z < x.Itfollows asily hat2z > x so thatS* is an archimedean xtension fS.Theconverses false.We givetwoexamples, hefirst fwhich s notdeep.(a) Let S be thegroupof ntegers. henS is d-complete; utthegroupof realsis an archimedeanxtension f S so thatS is nota-complete.ndeed,everydiscreteabeliantotally rderedgroup s d-complete ut not a-complete.(b) Let S be the abeliantotally rdered roupof sequencesofintegersrderedlexicographically.henf < g iff(n) < g(n), wheren is thefirst ositiventeger orwhichf(n)Ag(n). Iffor ach nwe takehj(n) = -1 andh,(m)= 0 form=# then hn,}is a sequenceofpositive lementswhich onverges o 0. It follows hat sequence{f,,}nS is a Cauchy equence f ndonly ffor ach N theres an nsuchthatr, > nimplies ,(k)= f,(k)for ach k< N.It is now an easy matter o see thatS is d-complete. or let {fl} be a Cauchy

sequence.Then foreach k define (k) = fr(k)wherer is so largethat ,n(k) fr(k)whenever ? r. We see that ,convergesof.On theotherhand,S is not a-complete ince thegroupT of sequencesof realnumbers,rderedexicographically,s an archimedeanxtension fS.In order o be able to state the theorem elating -completeness itha-com-pleteness,we needthenotionof convex ubgroup.

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8 MATHEMATICSAGAZINE [Jan.-Feb.Let S be an abelian totally rdered roup.A subgroup c S is called convexfx E I, x > 0, yE S, 0 < y < x, impliesyE I. Then is simplyn intervalnS centeredat 0. Let Ix and Iybe cosetsof . If Ix =, Yand x < y thenforeach uE Ix, vE IYit

easilyfollows hatu < v.We accordingly avean orderrelation etween hecosetsof I and it follows hatthe quotient roupS/I is an abeliantotally rdered roup.THEOREM5. An abelian totallyordered roup s a-completefand only if,foreach convex ubgroup of S, thequotient roupS/I is nondiscretend d-comnplete.The proof s noteasyand is given n [3].We aregratefulo R. 0. Daviesfor ismany emarkseadingo the mprovementf he oncep-tualstructurendpresentationf this rticle.

References1. G. BirkhoffndS. MacLane,A Survey f ModernAlgebra, rd d., Macmillan, ewYork,1965,p. 92.2. L. W. Cohen nd CasperGoffman,he topology fordered beliangroups, rans.Amer.Math. Soc., 67 (1949)310-319.3. L. W. Cohen nd CasperGoffman,n completenessnthe enseof Archimedes, mer.J.Math.,72 1950)747-751.4. CasperGoffman, eal Functions, inehart, ewYork, 1953, p. 122-134.5. H. Hahn,Uberdie nichtarchimedischenrossensysteme,itz.derKais. Akad.derWiss.,

Vienna, ect. Ila, 116 (1907) 601-653.6. F. Hausdorff,rundzuigeerMengenlehre,helsea,NewYork,1917, p. 194-207.7. D. Hilbert, rundlagenerGeometrie,eipzig, 903.8. 0. F. G. Schilling,he Theory fValuations, mer.Math. oc.,NewYork,1950, p. 6-7.9. Dana Scott,On completingrdered ields,n Applicationsf ModelTheory o Algebra,Analysis ndProbability,d. byW. A. J.Luxemburg,ewYork,1969, p. 274-278.

CARDAN'S FORMULAS AND BIQUADRATIC EQUATIONSROGER CHALKLEY, UniversityfCincinnati

1. Introduction.n 1956, Dr. Chao-Hui Yang showed me an interesting ayto use a 3 x 3 cyclicmatrix o recallCardan's formulas ortherootsof a cubicequation.Recently, yusing 4 x 4 cyclicmatrix n a similarmanner, discoveredanalogousformulas or herootsof biquadratic quation.Allthedetailsfor cubicare given n Section 2. Myresults ora biquadratic re presentedn Theorem ofSection4; they re related o other olution echniques n Sections5 and 6.As coefficientomain,suppose F is a field of characteristic 2,3 withtheproperty: or each elementy n F, X2 = yhas a solution n F and X3 = yhas asolution n F. In particular,he quadraticformulas applicable, and each second-degreepolynomial ver F has a root n F; thus,F contains principal ube rootco