An intuitionistic exploration of Zermelo’s Axiom

8/7/2019 An intuitionistic exploration of Zermelos Axiom

1/94

Whb romht ihdoib4

Cl oltuotodlostoi bxpjdrctodl df [brnbjds Cxodn

Godl Iduncls

Kujy 5118

Yupbrvosdr; gr. ].H.N. VbjgnclYbidlg rbcgbr; prdf. gr. C.I.N. vcl Uddok

Fciujty df YioblibUcgedug Rlovbrsoty Loknbmbl

Ytugblt lunebr; 19371??


2/94

Idltblts

Jost df Yynedjs 7

3 Oltrdguitodl 2

3.3 Hostdry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Dvbrvobw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =

5 Qrbrbquosotbs ?

5.3 Ecsoi sbts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.5 Ypbiocj suesbts dfL . . . . . . . . . . . . . . . . . . . . . . . . . 335.9 Yprbcgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.7 Erduwbrs Idltoluoty Qroliopjb . . . . . . . . . . . . . . . . . . . 395.2 Cxodns df Ihdoib . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.6 Fcls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3=

5.6.3 Fcl Whbdrbn . . . . . . . . . . . . . . . . . . . . . . . . . 3=

5.6.5 Rlrbstroitbg Fcl Whbdrbn . . . . . . . . . . . . . . . . . . 3?5.6.9 Bxtblgbg Fcl Whbdrbn . . . . . . . . . . . . . . . . . . . 53

5.= Whb sbt df thb rbcj lunebrs . . . . . . . . . . . . . . . . . . . . . 55

9 Gosirbtb sprbcgs clg fcls 52

9.3 Gosirbtb sprbcgs . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.5 Gosirbtb fcls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Dl dpbl, ijdsbg clg blunbrcejb suesbts dfL 937.3 Cjj `olgs df ijdsbg clg dpbl . . . . . . . . . . . . . . . . . . . . . 937.5 Ybqubltocjjy ijdsbg suesbts dfL . . . . . . . . . . . . . . . . . . 96

7.5.3 Folotbjy blunbrcejb suesbts dfL . . . . . . . . . . . . . . 967.5.5 Blunbrcejb suesbts dfL . . . . . . . . . . . . . . . . . . 9?

2 Dl thb bxostblib clg ldl-bxostblib df ibrtcol ihdoib fulitodls 76

2.3 Ihdoib fulitodls dl idjjbitodls df suesbts dfL . . . . . . . . . . . 762.5 Ihdoib fulitodls dl idjjbitodls df suesbts dfI . . . . . . . . . . . 7?

2.5.3 Folotbjy blunbrcejb suesbts dfI . . . . . . . . . . . . . . 212.5.5 Ybqubltocjjy ijdsbg suesbts dfI . . . . . . . . . . . . . . . 252.5.9 Bfibitovbjy dpbl suesbts dfI . . . . . . . . . . . . . . . . 27

5


3/94

6 Ybts df rbprbsbltctovbs fdr bquovcjblib rbjctodls 26

6.3 Bquovcjblib rbjctodls dlL . . . . . . . . . . . . . . . . . . . . . . 2=

6.5 Whb prosdlbrs clg thbor hcts . . . . . . . . . . . . . . . . . . . . . 656.5.3 Bquovcjblib rbjctodl dlI . . . . . . . . . . . . . . . . . . 656.5.5 Idllbitodl woth thb Votcjo rbjctodl . . . . . . . . . . . . . 66

6.9 Bquovcjblib rbjctodls clg ihdoib fulitodls . . . . . . . . . . . . . =16.9.3 Bquovcjblib rbjctodl movbl ey thb olhceotbgjy blunbrcejb

suesbts dfL . . . . . . . . . . . . . . . . . . . . . . . . . . =1

= Hcnbj ecsos ==

=.3 C Hcnbj olgbpblgblt fulitodl frdnI td U . . . . . . . . . . . . ===.5 C Hcnbj idnpjbtb fulitodl frdnI td U4 . . . . . . . . . . . . . 83=.9 Idllbitodl woth thb Votcjo rbjctodl . . . . . . . . . . . . . . . . . 89

=.9.3 Idlstruitolm wbc`jy Votcjo olgbpblgblt fulitodls . . . . . 87=.9.5 ]bc jy Votcjo olgbpblgblib dfmh . . . . . . . . . . . . . . 8==.9.9 Olidnpjbtblbss df wbc`jy Hcnbj olgbpblgblt fulitodls . 8?

=.7 Furthbr rbsbcrih . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9

Ubfbrblibs ?7

9


4/94

Jost df Yynedjs

CI1,1 Forst Cxodn df Idultcejb Ihdoib, 32IB suesbt dfL id-blunbrctbg ey , 33G suesbt dfL gbiogbg ey , 33B suesbt dfL blunbrctbg ey , 33MCI3,1 Forst Cxodn df Idltoludus Ihdoib, 36MCI3,3 Ybidlg Cxodn df Idltoludus Ihdoib, 36MIQ Erduwbrs Mblbrcjozbg Idltoluoty Qroliopjb, 37OB suesbt dfL olhceotbgjy blunbrctbg ey , 7=Y suiibssdr fulitodl, 75l suesbqublib df, 37# cpcrtlbss rbjctodl

ebtwbbl rbcj lunebrs clg oltbrvcjs, 6?dl Ecorb spcib, ?dl thb sbt df thb rbcj lunebrs, 59

rbjctodl dl thb idjjbitodl df oltbrvcjs, 59 idlictblctodl fulitodl, 35 idnpdsotodl fulitodl, 75 eokbitovb fulitodl frdn L tdL, 35L sbt df cjj ffilotb sbqublibs df lcturcj lunebrs, 35I Icltdr spcib, 39L Ecorb spcib, ?C ijdsurb df thb sbtC, 96n olotocj sbmnblt df df jblmth n, 35 olidnpctoeojoty rbjctodl, 75

c// strdlmjy ldt

c rbjctbg, 67

c bquovcjblib rbjctodl dlI, 65 drgbrolm rbjctodl

dl L, 35dl thb idjjbitodl df oltbrvcjs, 59

n olffilotb sbqublib woth idlstclt vcjubn, ?jm jblmth fulitodl

dl L, 35dl thb idjjbitodl df oltbrvcjs, 59

7


5/94

Ihcptbr 3

Oltrdguitodl

Oncmolb c qubub df olffilotbjy ncly prosdlbrs lunebrbg 1, 3, 5, . . .. Uclgdnjy,bcih df thbn os cssomlbg c ejci` dr c whotb hct. Bcih prosdlbr icl dljy sbb thbhcts df thb fbjjdw olnctbs ol frdlt df hon (o.b. thb hcts df thb olnctbs whd hcvbc homhbr lunebr thcl hb hcs). Whb mucrg cs`s bcih prosdlbr ol turl td mubssthb idjdur df hos hct, wothdut thb dthbr prosdlbrs ebolm cejb td hbcr hos rbpjy.Of thb prosdlbr clswbrs idrrbitjy, hb wojj eb rbjbcsbg. Of ldt, hb hcs td stcy olprosdl fdr thb rbst df hos jofb.Cftbr ebolm movbl thb rujbs df thb mcnb, thb prosdlbrs mbt dlb hdur td gbtbrnolbthbor strctbmy. Dlb df thbn, c ijcssoicj ncthbnctoiocl ciibptolm thb Cxodn dfIhdoib, scys O hcvb c pjcl thct blsurbs ct ndst ffilotbjy ncly df us mubsswrdlmjy.

Whb tdpoi df thos thbsos os thb Cxodn df Ihdoib, c stctbnblt whoih hcs jbgtd c humb cndult df idltrdvbrsy clg gosiussodl solib ots fdrnujctodl. Whb bx-cnpjb cedvb os kust dlb df thb ncly pbiujocr idlsbqublibs df thb Cxodn dfIhdoib. Eclcih clg Wcr`so bxpjcolbg, fdr bxcnpjb, hdw dlb, usolm thb Cxodn,icl iut c sphbrb oltd ffilotbjy ncly pobibs thct icl eb rbcrrclmbg oltd twd lbwsphbrbs thct crb edth df thb scnb sozb cs thb dromolcj dlb.

Ct ffirst somht thb Cxodn df Ihdoib sdulgs quotb pjcusoejb, cs ot stctbs;

fdr bvbry sbtC, thbrb bxosts c fulitodlf thct cssomls td bvbryldl-bnpty suesbtE dfC cl bjbnbltf(E) dfE.

(CI)

]b wojj rbfbr td thos stctbnblt cs CI.Fdr olstclib, fdr thb sbt C > {1, 3}, edth

{1} 1, {3} 3, {1, 3} 1,

clg

{1} 1, {3} 3, {1, 3} 3,

wdujg gd cs suih c fulitodl.Whos os c vbry sonpjb bxcnpjb. Ldtoib, CI stctbs fdr bvbry sbt, whoih nc`bsot c strdlm stctbnblt, ol pcrtoiujcr of dlb hcs td gbcj woth cl olffilotb sbt C.

2


6/94

Ol thos thbsos wb stugy thb Cxodn df Ihdoib frdn Erduwbrs oltuotodlostoi pdoltdf vobw. Whbrb os nuih idlfusodl cedut thb rdjb df thb Cxodn df Ihdoib ol

idlstruitovb ncthbnctois. Ydnb pbdpjb sbbn td rbmcrg ot cs c ldl-idlstruitovbproliopjb, whojb dthbrs gbfblg ot frdn c idlstruitovb pdolt df vobw. Whos idlfu-sodl os jcrmbjy icusbg ey thb ncly gofibrblt, ijcssoicjjy bquovcjblt, fdrnujctodlsdf thb Cxodn. Ct fcib vcjub, sdnb df thbsb fdrnujctodls sbbn pjcusoejb ol cidlstruitovb idltbxt, whojb dthbrs sbbn dutromht fcjsb.]b wojj shdw thct sdnb fdrns df thb Cxodn df Ihdoib crb ciibptcejb td thboltuotodlostoi ncthbnctoiocl, whojb dthbrs crb homhjy gbectcejb.Ebfdrb movolm c ndrb gbtcojbg gbsiroptodl df thb idltblt df thos thbsos, wb erobstys`btih thb dromol clg gbvbjdpnblt df thb Cxodn df Ihdoib. Ydnb fcnojocrotywoth ijcssoicj sbt thbdry os cssunbg, eut suih `ldwjbgmb os ldt rbquorbg fdr thbrbst df thos bsscy.

3.3 Hostdry

Ct thb blg df thb 3?th ibltury, Mbdrm Icltdr ijcons thct, fdr cjj sbts Xclg_,bothbrXicl eb bnebggbg oltd _ dr _ icl eb bnebggbg oltd X. Ol drgbr tdprdvb thos ijcon hb oltrdguibs thb ]bjj-Drgbrolm Qroliopjb, whoih stctbs thctbvbry sbt icl eb wbjj-drgbrbg. Hb prdpdsbs thos proliopjb cs c jcw df thdumht,c stctbnblt thct, ciidrgolm td hon, os devodus frdn thb wcy dlb shdujg thol`cedut sbts. Ldt bvbrydlb cmrbbs.Ol 3?17, sbcriholm fdr c prddf df thb ]bjj-Drgbrolm Qroliopjb, Brlst [brnbjdoltrdguibs thb Cxodn df Ihdoib.3 Whb fdrnujctodl df CI ey [brnbjd nc`bs

pbdpjb cwcrb df thb lunbrdus onpjoiot usbs df thos prdpdsotodl ol bcrjobr cr-munblts. Ot jbcgs td vbhbnblt gosiussodls; shdujg dlb ciibpt thb Cxodn dfIhdoib dr shdujglt dlb4Ol 3?18 [brnbjd puejoshbs cl cxodnctozctodl df sbt thbdry. Dlb df hos cons ostd ijcrofy thb rdjb clg nbclolm df CI. Furthbr stugy clg gbectb jbcgs td c jostdf lolb cxodns clg cxodn sihbnbs fdr sbt thbdry, icjjbg thb Cxodns df [brnbjdclg Frcbl`bj, ceerbvoctodl [F, whoih ffilgs wogb ciibptclib.Gub td thb wdr` df Frcbl`bj (3?55), Mdgbj (3?98), Ndstdws`o (3?72), clg Id-hbl (3?69) wb ldw `ldw thct thb Cxodn df Ihdoib os olgbpblgblt df [F, thctos, of [F os idlsostblt thbl edth [F+CI clg [F+CI crb idlsostblt. Idhblusbg c lbw nbthdg fdr hos prddf, icjjbg fdriolm, whoih turlbg dut td eb c vbry

fruotfuj nbthdg td detcol olgbpblgblib rbsujts P3^.

3

Whb Cxodn df Ihdoib os ol fcit bquovcjblt td thb ]bjj-Drgbrolm Qroliopjb.

6


7/94

3.5 Dvbrvobw

Whos thbsos os wrottbl frdn cl oltuotodlostoi pdolt df vobw. C ecsoi oltrdguitodltd oltuotodlosn os movbl ol Ihcptbr 5.Whb stctbnblt CI gdbs ldt nc`b onnbgoctb sblsb td thb oltuotodlostoi ncth-bnctoiocl. Whb idlibpt fdr bvbry sbt os fcr tdd mblbrcj clg wb prbfbr td ffirstidlsogbr sdnb spbiocj sbts C, jo`b thb sbt L df thb lcturcj lunebrs clg EcorbspcibL, td sbb whct ebidnbs df CI ol thdsb icsbs.Ydnb fdrns df thb Cxodn df Ihdoib crb ciibptcejb td us, cs thbsb fdjjdw frdnthb wcy wb thol` cedut thb dekbits df Ecorb spcib. ]b wojj bxpjcol clg kustofythdsb ol Ybitodl 5.2 clg wb wojj usb thbsb fdrns ol thb rbst df thb thbsos.Ol thb oltuotodlostoi stugy df sbts, thb bqucjoty rbjctodl gbsbrvbs spbiocj trbct-nblt. Dl thb sbt df thb lcturcj lunebrs, thb bqucjoty rbjctodl os gbiogcejb,eut dl Ecorb spcib ot os ldt. ]b hcvb trobg td ffilg c ihcrcitbrozctodl df edthsprbcgs clg fcls (twd typoicjjy oltuotodlostoi ldtodls, bxpjcolbg ol Ihcptbr 5)woth c gbiogcejb bqucjoty. Whb rbsujt os prbsbltbg ol Ihcptbr 9.

C sbidlg goiujty ol CI os hdw td ulgbrstclg thb phrcsb bvbry ldl-bnptysuesbt df C. Cmcol, thb ldtodl df cl creotrcry suesbt df c movbl sbt os tddmblbrcj. Ol oltuotodlostoi ncthbnctois ldt cjj suesbts crb df thb scnb òlg. Olthb bxcnpjb ct thb ebmollolm, wb hcvb dljy gbffilbg thb fulitodl fdr thb gb-iogcejb suesbts df{1, 3}. Whb stctbnblt ebidnbs strdlmbr of wb idlsogbr thbblunbrcejb suesbts dfC dr cl bvbl erdcgbr ldtodl df suesbt. ]b wclt ot td ebijbcr whoih idjjbitodl df (sue)sbts wb crb wdròlm woth, clg thbrbfdrb wb stcrtfrdn thb ijcssoicjjy bquovcjblt stctbnblt;

fdr bvbry fcnojyFdf ldl-bnpty sbts, thbrb bxosts c fulitodlf ;F

Fsuih thct, fdr bcihY F, f(Y) Y.

Movbl c fcnojy F, c fulitodl sctosfyolm thb cedvb idlgotodl os icjjbg c ihdoibfulitodl dl F. Olsporbg ey thos fdrnujctodl df thb Cxodn df Ihdoib, wb idl-sogbr vcrodus fcnojobs df ldl-bnpty sbts clg cs` whbthbr dlb icl gbffilb c ihdoibfulitodl dl thbn. Whos os thb tdpoi df Ihcptbr 2.Ebfdrb wb md oltd thb bxostblib clg ldl-bxostblib df ihdoib fulitodls, wb hcvbtd mcol c ebttbr olsomht oltd sdnb df thb fcnojobs df sbts wb wosh td idlsogbr,ol pcrtoiujcr thb idjjbitodl df thb ijdsbg suesbts dfLclg thb idjjbitodl df thbdpbl suesbts dfL. Cs wb bxpjcol ol Ihcptbr 7, thb oltuotodlostoi ncthbncto-

iocl hcs td gostolmuosh ebtwbbl sbvbrcj gbffilotodls df thb ldtodls dpbl suesbtdfL clg ijdsbg suesbt dfL, whoih wdujg eb bquovcjblt frdn c ijcssoicj pdoltdf vobw.

Ol ijcssoicj ncthbnctois, thb Cxodn df Ihdoib hcs c jcrmb lunebr df bquov-cjblt fdrnujctodls P5^. Ol thb jcst ihcptbrs wb stugy twd df thbsb clg sbb whctebidnbs df thbn of wb try td oltbrprbt thbn oltuotodlostoicjjy. Ol Ihcptbr 6 wbstcrt frdn thb stctbnblt;

bvbry bquovcjblib rbjctodl hcs c sbt df rbprbsbltctovbs.

]b try td ffilg dut of, fdr sbvbrcj òlgs df bquovcjblib rbjctodls dl thb lcturcj

lunebrs, thbrb bxosts c sbt df rbprbsbltctovbs clg ol whct sblsb.

=


8/94

]b cjsd rbturl td thb prdejbn df thb prosdlbrs clg thbor hcts fdrnujctbg ct thbebmollolm df thos oltrdguitodl clg bxpjcol thb ijcssoicj strctbmy df thb prosdlbr

thct ijcons td eb cejb td mbt ndst df thbn frbb. ]b wojj sbb thct hb usbs thbfcit thct thbrb bxosts c sbt df rbprbsbltctovbs fdr c ibrtcol bquovcjblib rbjctodldl Icltdr spcib I. Whos bquovcjblib rbjctodl cppbcrs td eb idllbitbg woth thbfcndus bquovcjblib rbjctodl gbffilbg ey Votcjo dl thb sbt U df thb rbcj lunebrstd idlstruit c ldl-Jbebsmub nbcsurcejb suesbt dfU. Idlsogbrolm thb rbjctodldlIfrdn cl oltuotodlostoi pdolt df vobw, wb prdvb sdnb thbdrbns td thb bfibitthct ot fcojs td hcvb cl bcsojy detcolcejb sbt df rbprbsbltctovbs.Ol Ihcptbr =, wb idlsogbr thb fdjjdwolm ijcssoicj bquovcjblt df CI;

bvbry vbitdr spcib hcs c ecsos.

]b stugy U cs c vbitdrspcib dvbr S clg olvbstomctb thb bxostblib df c ecsos

fdr U dvbr S, c sd-icjjbg Hcnbj ecsos. ]b shdw thct, cs ol ijcssoicj sbt thbdry,thbrb os c ijdsb idllbitodl ebtwbbl Hcnbj ecsbs clg Votcjo sbts. Rsolm thosidllbitodl wb prdvb sdnb pdsotovb rbsujts shdwolm thbrb os ld bcsojy detcolcejbHcnbj ecsos.

8


9/94

Ihcptbr 5

Qrbrbquosotbs

Whos ihcptbr prdvogbs c shdrt oltrdguitodl td oltuotodlostoi ncthbnctois. Ydnbonpdrtclt ecsoi idlibpts df oltuotodlosn crb bxpjcolbg clg O hdpb thct ey mdolmthrdumh thb bxcnpjbs thb rbcgbr mbts sdnb fbbjolm fdr c lunebr df typoicjjyoltuotodlostoi proliopjbs. Gbffilotodls df ndrb spbioffii idlibpts usbg ol thos thbsoswojj eb movbl cjdlm thb wcy.

5.3 Ecsoi sbts

]b stugy twd ecsoi sbts. Whb ffirst os L, thb sbt df thb lcturcj lunebrs. Lhcs c sd-icjjbg gbiogcejb bqucjoty. Whos nbcls wb icl gbiogb fdr cjjn, l ol Lwhbthbrn > l drn >l.

Whos os ldt sd fdr thb sbidlg fulgcnbltcj sbt, L, thb idjjbitodl df cjj olffilotbsbqublibs df lcturcj lunebrs, dr, dlb idujg scy, thb idjjbitodl df cjj fulitodlsfrdn L td L. L os icjjbg Ecorb spcib. Whb bjbnblts dfL crb thdumht df csebolm idlstruitbg stbp ey stbp. Dlb gbffilbs cl bjbnblt dfLey movolm, dlbey dlb, ots vcjubs (1), (3), (5), . . .. Yuih c sbqublib ncy eb movbl ey clbxpjoiot cjmdrothn (fdr bxcnpjb poi` 1 ol bcih stbp), eut, bvbl thbl, ot nomhteb oncmolbg td eb thb rbsujt df ihddsolm rclgdnjy ol bcih stbp.

Gbffilotodl 5.3. Fdr bcih n L, wb gbffilb cl bjbnblt n dfL ey

n(l) >g

n fdr cjjl ol L.

Fdr bxcnpjb, 1 os thb sbqublib idlsostolm df dljy zbrds.

Fdr , L, wb gbffilb

> l P (l) >(l) ^.

Ycyolm ( > ) os scyolm thct thb cssunptodl l P (l) > (l)^ jbcgs tdc idltrcgoitodl. Whos os quotb wbc . ]b dljy `ldw thct clg crb ldtbvbrywhbrb thb scnb, eut wb ncy ldt eb cejb td olgoictb c pjcib whbrb thbygofibr. Whbrbfdrb, wb cjsd gbffilb c strdlmbr rbjctodl dl L.

Gbffilotodl 5.5. ]b scy; os cpcrt frdn, ldtctodl # , of clg dljy of thbrb

bxostsl L suih thct(l)>(l).

?


10/94

Ol oltuotodlostoi ncthbnctois thb phrcsb thbrb bxosts os oltbrprbtbg ol c idl-struitovb wcy. Whct os, fdr cly prdpbrty C df lcturcj lunebrs, thb stctbnblt

l L P C(l) ^ nbcls thct wb crb cejb td bxpjoiotjy ffilg c lcturcj lunebr l1sctosfyolmC(l1).

]b iclldt gbiogb fdr bcih , L,

> #.

Jbt us ffirst bxpjcol hdw wb, fdjjdwolm Erduwbr, oltbrprbt thb jdmoicj synedj .Fdr cly twd ncthbnctoicj stctbnblts C clg E ,C E nbcls thct wb icl poi`dlb df thb twd stctbnblts (C dr E ) clg movb c prddf df thos stctbnblt.Ldw idlsogbr fdr olstclib L gbffilbg ey;

(l) >g

1 of thbrb os ld ejdi` df ?? lolbs ol thb ffirstl gomotsdf thb gbioncj bxpclsodl df,

>g

3 of thbrb os c ejdi df ?? lolbs ol thb ffirstl gomots df

thb gbioncj bxpclsodl df.

Ldtb thct

l P (l) > 1^ thbrb os ld ejdi df ?? lolbs ol thb gbioncj bxpclsodl df,

clg

l P (l)> 1 ^ thbrb os c ejdi` df ?? lolbs ol thb gbioncj bxpclsodl df.

Cs, ct thos ndnblt, wb gd ldt `ldw whbthbr dr ldt thbrb bxosts c ejdi` df ??lolbs ol thb gbioncj bxpclsodl df , wb iclldt prdvb > 1, ldr # 1. Hblibwb iclldt gbiogb > 1 # 1.

]b wojj usb thb bxcnpjb df thb ?? lolbs ol thb gbioncj bxpclsodl df ndrb df-tbl, clg thbrbfdrb, wb oltrdguib sdnb ceerbvoctodls. Fdr bcih lcturcj lunebrl wb gbffilb

l : `?? >g

thbrb os ld ejdi` df ?? lolbs ol thb ffirst l gomots df thb gbioncj

bxpclsodl df.l> `?? >

g

l os thb ffirst pdsotodl ol thb gbioncj bxpclsodl df whbrb c ejdi`

df ?? lolbs os idnpjbtbg.

l < `?? >g

thbrb os c ejdi` df ?? lolbs ol thb ffirst l 3 gomots df thb gbioncj

bxpclsodl df.

Ldtb thct wb gd ldt movb c gbffilotodl df `?? dl ots dwl. Lbvbrthbjbss, wbwojj diicsodlcjjy tcj` cedut `?? cs of ot bxosts ol ots dwl romht. ]b idujg, fdrbxcnpjb, scy `?? bxosts, whbl wb nbcl; l P l> `??^.

31


11/94

5.5 Ypbiocj suesbts dfL

JbtC eb c suesbt dfL clg l c lcturcj lunebr. ]b iclldt gbiogb ol mblbrcj

l C l /C. (5.3)

Dlb idujg gbffilb, fdr bxcnpjb,

C>g

{l L | l> 1 thb Uobncll Hypdthbsos}.

Cs wb icl lbothbr prdvb ldr rbkbit thb Uobncll Hypdthbsos (ct thos ndnblt),wb iclldt gbiogb 1 C 1 /C.C suesbt C dfL fdr whoih wb crb cejb td gbiogb (5.3) fdr cjj lcturcj lunebrsl os icjjbg c gbiogcejb suesbt dfL. Ol fcit, wb wojj usb thb fdjjdwolm gbffilotodldf thb ldtodl gbiogcejb suesbt dfL, whoih os sjomhtjy strdlmbr.

Gbffilotodl 5.9. Fdr cly L, wb gbffilb;

G>g

{l L | (l) > 3}.

JbtX L.]b scy thct gbiogbs Xof clg dljy ofX idoliogbs woth G.X os c gbiogcejb suesbt dfL of clg dljy of sdnb L gbiogbs X.

Wwd dthbr spbiocj idjjbitodls df suesbts dfL crb thb idjjbitodl df thb blunbrcejbsuesbts dfL clg thb idjjbitodl df thb id-blunbrcejb suesbts dfL. Whbsb crbgbffilbg cs fdjjdws.

Gbffilotodl 5.7. Fdr cly L, wb gbffilb;

B>g

{l L | ` L P l> (`) 3 ^},

IB>g

L Z B> {l L | (l B) }.

JbtX L.]b scy thct blunbrctbs (rbsp. id-blunbrctbs) Xof clg dljy ofXidoliogbswoth B (IB).X os cblunbrcejb suesbt (rbsp. id-blunbrcejb suesbt) dfL of clg dljy of sdnb Lblunbrctbs (id-blunbrctbs) X.

Ldt bvbry blunbrcejb suesbt dfL os gbiogcejb. Fdr olstclib, thb sbt

C >g

{l L | l> 3 `??bxosts}

os ldt gbiogcejb. Cs wb gd ldt `ldw whbthbr dr ldt thbrb bxosts c ejdi df?? lolbs ol thb gbioncj bxpclsodl df, wb crb ldt cejb td gbiogb whbthbr thblcturcj lunebr 3 os ol C dr ldt.Cos blunbrcejb fdr wb icl gbffilb L ey

(`) >g

1 of : `??,

>g

5 of `??.

WhblC > B.

]b wojj scy sdnb ndrb cedut thos tdpoi ol Ybitodl 6.9.3.

33


12/94

5.9 Yprbcgs

C typoicj oltuotodlostoi idlibpt os thb ldtodl df c sprbcg. Ol drgbr td gbffilbsprbcgs wb ffirst oltrdguib sdnb dthbr idlibpts. Wd stcrt woth, L os thb sbt df cjjffilotb sbqublibs df lcturcj lunebrs. ]b gbffilb c eokbitovb fulitodl ; L Ley gbffilolm, fdr cjj ` L, fdr cjj s1, . . . , s`3 L,

s1, . . . , s`3 >g

ps11 . . . ps`5`5 p

s`3+3`3 3 (5.5)

whbrb pl os thb lth pronb lunebr clg ey gbffilotodl thb bnpty prdguit os 3

(whblib > 3 3 > 1).Rsolm thos fulitodl, wb ncy vobw L clg L cs thb scnb sbt.]b gbffilb c fulitodl jm ; L L ey

jm( s1, . . . , s`3 ) >g `.

]b cjsd gbffilb, fdr bcih ffilotb sbqublib s clg cjj o : jm(s),

s1, . . . , s`3(o) >g

so.

]b gbffilb twd idlictblctodl fulitodls, edth gbldtbg ey . Whb ffirst dlb osebtwbbl ffilotb sbqublibs clg thb sbidlg dlb os ebtwbbl ffilotb sbqublibs clgolffilotb sbqublibs.

Gbffilotodl 5.2. Jbt L clgr, s L, scyr > r1, . . . rn,s > s1, . . . , sl.

r s>g

r1, . . . , rn, s1, . . . , sl,

r >g

(r1, . . . , rn, (1), (3), (5), . . .).

Whb idjjbitodl df cjj ffilotb sbqublibs df lcturcj lunebrs ncy eb drgbrbg csfdjjdws.

Gbffilotodl 5.6. Jbtc, e L.]b scy c bxtblgs td e, ldtctodl c e, of clg dljy of

iL P e> c i ^.

]b scy c os c prdpbr olotocj sbmnblt dfe, ldtctodl c e, of clg dljy of

c e c>e.

Gbffilotodl 5.=. Fdr cjj L, fdr cjj n L,

(n) >g

(1), . . . , (n 3).

Ol icsb thbrb os ld idlfusodl pdssoejb, wb jbcvb dut thb pcrblthbsbs clg wrotbn. ]b icjj n cl olotocj sbmnblt df.

Yonojcry, fdr cjj s L, fdr cjj n jm(s),

s(n) >g

s(1), . . . , s(n 3) .

35


13/94

Ldw wb crb rbcgy td gbffilb thb ldtodl df c sprbcg.

Gbffilotodl 5.8. Jbt ; L {1, 3}.

]b scy os csprbcg jcw of clg dljy of;

(J3) ( ) > 1,

(J5) fdr cjj s L, (s) > 1 l P (s l ) > 1^.

Fdr cjj L, wb scy; os cgnottbg ey, ldtctodl , of clg dljy of fdr cjjl L, (l) > 1. Whb sbt df cjj sbqublibs thct crb cgnottbg ey c sprbcg jcw os icjjbg c sprbcg. ]b wrotb fdr edth thb sprbcg clg thb sprbcg jcw.

Dlb icl thol` dfL cs c trbb woth thb ffilotb sbqublibs df lcturcj lunebrs csldgbs. Ol thos trbb thbrb os c pcth frdn thb ffilotb sbqublibc td thb ffilotb sb-

qublib e of clg dljy ofc bxtblgs td e. Whb bjbnblts dfL crb cjj thb olffilotbpcths dlb ncy idlstruit ey mdolm upwcrgs ol thos trbb.Fdr c movbl sprbcg , thb sprbcg jcw tbjjs ydu, ol bcih ldgb, whoih erclihbsydu icl wcj` of ydu wclt td idlstruit cl olffilotb sbqublib thct os cl bjbnbltdf. Gub td prdpbrty J5 thbrb crb ld gbcg blgs. Qrdpbrty J3 mucrcltbbs thct idltcols ct jbcst dlb bjbnblt.

Icltdr spcib, I, os thb idjjbitodl df cjj olffilotb sbqublibs df lcturcj lunebrscssunolm ld dthbr vcjubs thcl zbrd clg dlb. Whos os c sprbcg, cs wb icl gbffilbthb cpprdproctb sprbcg jcw ey;

(s) > 1 o : jm(s) P s(o) > 1 s(o) > 3^

5.7 Erduwbrs Idltoluoty Qroliopjb

JbtU L L eb c rbjctodl suih thct, fdr cjj olL, wb icl ffilg c lcturcjlunebr l sctosfyolmU(, l). Whbl Erduwbrs Idltoluoty Qroliopjb (IQ) stctbsthct, fdr cjj ol L, wb icl cjrbcgy gbtbrnolb c lcturcj lunebr l sctosfyolmU(, l) cftbr `ldwolm dljy c ffilotb pcrt df, o.b.

L n L l L LP n> n U(, l) ^

Whos nomht sdulg onprdecejb ct ffirst, eut rbicjj hdw wb vobw bjbnblts df Ecorbspcib. Bvbry olffilotb sbqublib df lcturcj lunebrs ncy eb thdumht df cs ebolmidlstruitbg stbp ey stbp. Of ydu stctb

L l L P U(, l) ^,

ydu ijcon thct of sdnbdlb (jbt us icjj thct pbrsdl ydur dppdlblt) movbs yducl bjbnblt dfLey movolm ydu dlb ey dlb ots vcjubs (1), (3), (5), . . ., ctsdnb pdolt ydu crb cejb td idnb up woth c lcturcj lunebrl suih thctU(, l).Yuppdsb ydu gd sd cftbr ydur dppdlblt hcs movbl thb ffirst 311 vcjubs df. Whblfdr cly thct cmrbbs dl thb ffirst 311 pdsotodls woth (o.b. 311 >311), cjsdU(, l), cs ydu gd ldt ybt `ldw clytholm cedut thb futurb vcjubs df.

39


14/94

Ol Ybitodl 5.3 wb hcvb idlstruitbg c sbqublib fdr whoih thb stctbnblt

> 1 # 1 os rbi jbss. Rsolm thb Idltoluoty Qroliopjb dlb icl detcolc idltrcgoitodl frdn thb cssunptodl , LP > # ^. Forst ldtbthct wb icl idgb c sbqublib df bjbnblts 1, 3, . . . dfLey kust dlb bjbnbltdfL. Wd gd sd wb usb thb ncp gbffilbg ct (5.5) dl pcmb 35.

Gbffilotodl 5.?. Jbt L.Fdr bcih l L, wb gbffilb l Ley, fdr cjj n L,

l(n) >g

( l, n ).

Ldw suppdsb

, LP > # ^. (5.9)

Gbffilb U L L eyU(, 1) 1 >3 U(, 3) 1 #3.

Ey (5.9), ol pcrtoiujcr LP 1 >3 1 #3 ^, sd

L l L P U(, l) ^.

Cppjyolm thb Idltoluoty Qroliopjb yobjgs;

L n L l L LP n> n U(, l) ^. (5.7)

Ldtb thct 11 > 1 > 13, sdU(1, 1) clg ldt U(1, 3).Hblib, ey cppjyolm (5.7) fdr > 1, wb icl gbtbrnolb n L suih thct;

LP 1n> n U(, 1) ^. (5.2)Gbffilb L ey

(l) >g

1 ofl >3, n,

>g

3 ofl > 3, n.

Whbl, cs 3, n< n, n> 1n. Hblib, ey (5.2), U(, 1).Eut1 > 1>3. Idltrcgoitodl.Yd, usolm thb Idltoluoty Qroliopjb, wb hcvb prdvbl

, LP > # ^.

Ol Ybitodl 5.9 wb hcvb oltrdguibg sprbcgs. Whb crmunblt movbl td kustofy thbIdltoluoty Qroliopjb fdr Licl eb cppjobg td cly sprbcg.

Gbffilotodl 5.31. Erduwbrs Mblbrcjozbg Idltoluoty Qroliopjb (MIQ).Jbt Leb c sprbcg clg U L.Of, fdr cjj ol , thbrb bxosts c lcturcj lunebr l sctosfyolm U(, l), thbl

n L l L P n> n U(, l) ^.

Whb (Mblbrcjozbg) Idltoluoty Qroliopjb os c pdwbrfuj tddj. Rsolm MIQ dlb iclfdr bxcnpjb prdvb thct bvbry rbcj fulitodl os idltoludus (thb gbffilotodl df thbrbcj lunebrs wojj eb movbl ol Ybitodl 5.=). Whb Idltoluoty Qroliopjb wojj pjcy

cl onpdrtclt rdjb ol ncly prddfs ol thos thbsos.

37


15/94

5.2 Cxodns df Ihdoib

Ol thos thbsos wb wojj olvbstomctb whct ebidnbs df thb vcrodus fdrns clg bquovc-jblts df thb Cxodn df Ihdoib of wb try td oltbrprbt thbn oltuotodlostoicjjy. Ydnbfdrns df thb Cxodn df Ihdoib crb ciibptcejb fdr thb oltuotodlostoi ncthbncto-iocl, cs thbsb fdjjdw frdn thb wcy wb thol` cedut bjbnblts df Ecorb spcib. Whbffirst df thbsb os thb Forst Cxodn df Idultcejb Ihdoib

Gbffilotodl 5.33. Forst Cxodn df Idultcejb Ihdoib (CI1,1)JbtQ L L suih thct, fdr bcih lcturcj lunebr l, wb icl ffilg ct jbcst dlbn ol L woth Q(l, n). Whbl thbrb bxosts L suih thct l L P Q(l, (l))^.

Ol ijcssoicj sbt thbdry CI1,1 os c thbdrbn (o.b. c stctbnblt dlb icl prdvb frdn[F, thb usucj cxodn systbn fdr sbt thbdry wothdut thb Cxodn df Ihdoib), cs

dlb ncy gbffilb c fulitodl f ; L L, ey

f(l) >g

n P Q(l, n) ^.

Oltuotodlostoicjjy, thos fdrnujc gdbs ldt gbffilb cl bjbnblt dfL, cs wb ncy ldteb cejb td gbiogb, fdr c movbl l, whoih nL os thb sncjjbst lcturcj lunebrsctosfyolmQ(l, n).Hdwbvbr, CI1,1 os ciibptbg cs cl cxodn, ebicusb ot fdjjdws frdn thb wcy wbvobw L. Whb bjbnblts dfL crb gbffilbg stbp ey stbp. Hblib fdr cly suesbtQdfL L woth

nLl L P Q(n, l) ^,

wb icl gbffilb cl bjbnblt cs fdjjdws; ol stbp l ffilg n L woth Q(l, n) clggbffilb (l) >

g

n. Whos sctosffibs thb rbquorbnblt.

Whb Forst Cxodn df Idultcejb Ihdoib os cedut rbjctodls dl L. Whb lbxt ihdoibcxodn wb idlsogbr os cedut rbjctodls ebtwbbl L clg L. Ebfdrb wb icl fdrnu-jctb thos cxodn wb lbbg td eb ijbcr dl whct wb nbcl ey c fulitodl frdn L tdL. ]b onnbgoctbjy gbffilb c ndrb mblbrcj ldtodl, lcnbjy thb dlb df c fulitodlfrdn c sprbcg td thb lcturcj lunebrs.

Gbffilotodl 5.35. Jbt , L. ]b gbffilb

;l >g

cssomls td thb lcturcj lunebrl,

>g

` L P (`) >l + 3 nL P n>` (n) > 1 ^ ^.

Fdr cly sprbcg clg cly L wb gbffilb

; L >g

gbffilbs c fulitodl frdn td L,

>g

l L P ; l ^.

Fdr cjj , wb wrotb() fdr thb uloqub lcturcj lunebrl suih thct;

l.

32


16/94

]hy gdbs ot nc`b sblsb td gbffilb fulitodls frdn c sprbcg td L jo`b thos4]bjj, suih c fulitodl sblgs bcih bjbnblt df td c lcturcj lunebr. Of sdnbdlb

os euojgolm cl bjbnblt df thb sprbcg ey movolm ots vcjubs stbp ey stbp, ct sdnbpdolt thb fulitodl shdujg eb cejb td suppjy thb oncmb df thos sbqublib. Whos osbxcitjy whct hcppbls woth thb gbffilbg cedvb. Fdr cly L, fdr cjjn L,cs jdlm cs thb olotocj sbmnblt n df gdbs ldt movb bldumh olfdrnctodl tdgbtbrnolb thb oncmb df, (n) > 1. Hdwbvbr, ct sdnb pdolt wb `ldw whctthb oncmb df os, scy cftbr `ldwolm thb ffirst ` vcjubs df wb `ldw ots oncmbosl. Whbl (`) >l + 3.

Ldw wb crb rbcgy td gbffilb thb lbxt ihdoib cxodn, thb Forst Cxodn df Idl-toludus Ihdoib.

Gbffilotodl 5.39. Forst Cxodn df Idltoludus Ihdoib (MCI3,1)

Jbt eb c sprbcg clg U L suih thct, fdr cjj , thbrb bxosts l LwothU(, l). Whbl thbrb bxosts c fulitodl ; L, suih thct, fdr cjj L,U(, ()).

Whos cxodn ncy eb gbfblgbg cs fdjjdws. Jbt eb c sprbcg clg U Lsuih thct l L P U(, l) ^. ]b gbffilb L stbp ey stbp. Fdr bcihc L gbiogb whbthbr c movbs bldumh olfdrnctodl td ffilg l L woth U(, l)fdr c sbqublib stcrtolm woth (thb ffilotb sbqublib idgbg ey) c. Of thos os thbicsb clg thbrb os ld e L woth e c clg (e) > 1, wb gbffilb (c) >

g

l+ 3.

Dthbrwosb, (c) >g

1. Dlb ncy idlvolib dlbsbjf thct os c fulitodl frdn td

L clg, fdr cjj , U(, ()).

Whb jcst ihdoib cxodn wb oltrdguib os cedut rbjctodls dl L. ]b ffirst gbffilbwhct wb nbcl ey c fulitodl frdn c sprbcg td L.

Gbffilotodl 5.37. Jbt,, L. ]b gbffilb;

; >g

cssomls td thb sbqublib,

>g

l L P l ;(l) ^.

Fdr cly sprbcg clg L wb gbffilb;

; L >g

gbffilbs c fulitodl frdn td L,

>g

LP ; ^.

Fdr cjj , wb wrotb | fdr thb uloqub sbqublib suih thct ; .

Gbffilotodl 5.32. Ybidlg Cxodn df Idltoludus Ihdoib (MCI3,3)Jbt eb c sprbcg clg U L suih thct, fdr cjj , thbrb bxosts LwothU(, ). Whbl thbrb bxosts c fulitodl ; Lsuih thct, fdr cjj L,U(, |).

36


17/94

Ot tc`bs c eot ndrb bfidrt td gbfblg thos cxodn. Fdr thb kustoffiictodl df thos cxodnclg c thdrdumh bxpjclctodl df cjj ihdoib cxodns oltrdguibg ol thos sbitodl thb

rbcgbr os rbfbrrbg td P7^.MCI3,3 os thb strdlmbst cxodn wb hcvb oltrdguibg ol thos sbitodl. EdthCI1,1clg MCI3,1 crb c idlsbqublib dfMCI3,3.

5.6 Fcls

]b hcvb sbbl twd bxcnpjbs df sprbcgs up td ldw; L clg I. Whbrb os clonpdrtclt gofibrblib ebtwbbl thbsb twd sprbcgs. ]hbl idlstruitolm cl bjbnbltdfLydu hcvb olffilotbjy ncly pdssoejb ihdoibs ct bcih stbp (ydu icl poi` clylcturcj lunebr). Dl thb dthbr hclg, ydu dljy hcvb twd dptodls (zbrd clg dlb)fdr bcih pdsotodl whbl ydu crb idlstruitolm cl bjbnblt dfI. C sprbcg ol whoih

ydu dljy hcvb ffilotbjy ncly pdssoejb ihdoibs ol bcih stbp os icjjbg c fcl. Cprbiosb gbffilotodl df thb ldtodl df c fcl os thb fdjjdwolm.

Gbffilotodl 5.36. Jbt ; L {1, 3}.]b scy os c fcl jcwof clg dljy of os c sprbcg jcw, o.b. sctosffibs;

(J3) ( ) > 1,

(J5) fdr cjj s L, (s) > 1 l P (s l ) > 1^,

clg ol cggotodl,

(J9) fdr cjj s L, (s) > 1 ` L l L P (s l) > 1 l : ` ^.

Fdr cly fcl jcw thb sbt

{ L |l L P (l) > 1 ^},

o.b. thb sbt df cjj olffilotb sbqublibs thct crb cgnottbg ey , os icjjbg c fcl.

Dlb ncy vobw c fcl cs c trbb, whbrb ol bcih ldgb thbrb crb ffilotbjy ncly(eut lbvbr zbrd) erclihbs mdolm up. Whb bjbnblts df c fcl crb idlstruitbg eywcj`olm up thb trbb clg ol bcih ldgb poi`olm dlb df thb ffilotbjy ncly pdssoejberclihbs td c fdjjdwolm ldgb.

5.6.3 Fcl Whbdrbn

Whb Fcl Whbdrbn os c pdwbrfuj tddj usbg ol oltuotodlostoi ncthbnctois. Erduwbrusbg thos thbdrbn td prdvb thct bvbry rbcj-vcjubg (idltoludus) fulitodl gbffilbgdl c ijdsbg oltbrvcj os ulofdrnjy idltoludus. Forst wb oltrdguib thb ldtodl df cecr.

Gbffilotodl 5.3=. Jbt eb c fcl clg E c suesbt dfL.]b icjj E c ecr ol of clg dljy of bvbry ol hcs cl olotocj sbmnblt lebjdlmolm td E .

3=


18/94

Whb Fcl Whbdrbn scys

bvbry gbiogcejb ecr hcs c ffilotb sueecr.

]hct gd wb nbcl ey thos4 Jbt eb c fcl clg E c gbiogcejb suesbt dfL

thct os c ecr ol . Whbl thb Fcl Whbdrbn stctbs wb icl ffilg c ffilotb suesbt E

dfE, suih thct E os c ecr ol . Dlb ncy fdrnujctb thb Fcl Whbdrbn ol thbfdjjdwolm bquovcjblt wcy cs wbjj.

Gbffilotodl 5.38. Fcl WhbdrbnFdr bvbry fcl clg fdr bvbry gbiogcejb suesbt E dfL thct os c ecr ol ,thbrb bxosts c lcturcj lunebrNsuih thct, fdr cjj , thbrb bxosts c lcturcjlunebr l N woth l E.

Ijcssoicjjy, thb Fcl Whbdrbn os bquovcjblt td ots idltrcpdsotodl, whoih os `ldwl

cs @dloms Jbnnc;

bvbry olffilotb ffilotbjy ercliholm trbb hcs cl olffilotb pcth.

Whbrb crb sdnb bxprbssodls ol thb stctbnblt cedvb thct pbrhcps hcvb td ebbxpjcolbg. Yd jbt us stcrt woth thct.

Gbffilotodl 5.3?. JbtW L.W os c trbb of clg dljy of fdr cjj s L, fdr cjj l L, ofs l ebjdlms td W,thbl s ebjdlms td W.C trbb W os ffilotbjy-ercliholm of clg dljy of fdr cjj bjbnblts s df W, thbrb crbdljy ffilotbjy ncly lcturcj lunebrs l woth s l W.C trbbW osolffilotbof clg dljy of fdr cjj l L, thbrb bxostss Wwothjm(s) >l.Cl bjbnblt dfL os cl olffilotb pcth dfWof clg dljy of fdr cjj l L, l W.

Whb ijcssoicj ncthbnctoicl ncy prdvb @dloms Jbnnc cs fdjjdws. Jbt W ebcl olffilotb ffilotbjy-ercliholm trbb. Whb ijcssoicj ncthbnctoiocl ldw idlstruitscl olffilotb pcth df W stbp ey stbp. Ol thb ffirst stbp hb ldtbs thct W osffilotbjy-ercliholm, hblib thbrb crb dljy ffilotbjy ncly l L woth l W, scyl1, . . . , l`. Cs cly bjbnblt dfWstcrts woth bothbr l1 dr l3 dr . . . dr l` clg Wos olffilotb, ct jbcst fdr dlb df thb l o thb sbt

Co >g

{s L | s(1) >lo }

os olffilotb. (1) os gbffilbg cs thb sncjjbst suih lo. Idltoluolm thos wcy, thbijcssoicj ncthbnctoiocl gbffilbs, fdr bcih l L,

(l) >g

` P {s W| l ` s}os olffilotb ^

Hb prdvbs ey olguitodl thct suih ` bxosts fdr bcih l, hblib os c wbjj-gbffilbgolffilotb pcth ol W. Yd ijcssoicjjy @dloms Jbnnc hdjgs clg thbrbey cjsd thbFcl Whbdrbn hdjgs.Whos prddf df @dloms Jbnnc os ldt ciibptcejb fdr thb oltuotodlostoi ncthb-nctoiocl. Hcvb cldthbr jdd` ct hdw (1) os gbffilbg. Cjthdumh thb ulodl df

thb sbts C1, . . . , C` os olffilotb, wb ncy ldt eb cejb td gbiogb whoih df thb sbts

38


19/94

os olffilotb. ]b dljy `ldw thct thby crb ldt cjj ffilotb. Whb scnb prdejbn crosbswhbl gbffilolm (3), (5), . . ..

Hcvolm sbbl thos, wb lbbg ldt eb suprosbg thct @ dloms Jbnnc gdbs ldt hdjgoltuotodlostoicjjy. Dlb idujg gbffilb c trbb W ey

s W l L P s> 1l (l : `?? (l `?? `??os bvbl) ^

l L P s> 3l (l : `?? (l `?? `?? os dgg)^.

Whbl W os cl olffilotb ffilotbjy-ercliholm trbb, eut ot os rbi`jbss td scy Whcs clolffilotb pcth. Fdr suppdsb Whcs cl olffilotb pcth, scy . Whbl

(1) > 1 l L P l> `?? los bvbl ^,

(1) > 3 l L P l> `?? los dgg ^.

Cs wb crb cejb td gbiogb whbthbr (1) > 1 dr (1) > 3, ot fdjjdws frdn thbbquovcjbliobs cedvb thct wb icl gbiogb

l L P l> `?? los bvbl ^ l L P l> `?? los dgg ^.

Eut thos os rbi`jbss.

Erduwbr prdvbg thb Fcl Whbdrbn usolm hos Qroliopjb df Ecr Olguitodl. Fdrc fujj bxpjclctodl df hos crmunblt thb rbcgbr idujg idlsujt P9^ dr P7^.

5.6.5 Rlrbstroitbg Fcl Whbdrbn

Rp td ldw wb hcvb dljy spd`bl cedut gbiogcejb ecrs. Jbcvolm dut thb cgkbitovb

gbiogcejb ol thb fdrnujctodl df thb Fcl Whbdrbn movbs us c ndrb mblbrcj fdrndf thos thbdrbn. Ldtb thct thos gdbs ldt ihclmb clytholm fdr thb ijcssoicjncthbnctoiocl. Hb cssunbs thb jcw df bxijugbg noggjb ey whoih bvbry ecr oscutdnctoicjjy c gbiogcejb ecr.

Gbffilotodl 5.51. Rlrbstroitbg Fcl WhbdrbnJbt eb c fcl clg jbt E L eb c ecr ol . Whbl wb icl ffilgL L suihthct l L P l L (, l) E ^.

Whbdrbn 5.53. CssunolmMCI3,1, thb Rlrbstroitbg Fcl Whbdrbn icl eb gb-rovbg frdn thb Fcl Whbdrbn.

Qrddf. Jbt eb c fcl clg E L

c ecr ol .]b wojj shdw thct E hcs c ffilotb sueecrE .Cs E os c ecr ol ,

l L P l E ^.

Yd, ey MCI3,1, thbrb bxosts c fulitodl ; L suih thct;

P (()) E ^.

Gbffilb df suesbtI dfL ey

I>g

{s L | (s)> 1}.

3?


20/94

Forst ldtb thct I os c ecr ol .

Jbt .

; L, sd ey gbffilotodl thbrb bxosts l L woth (l)> 1.

Yd l L P l I^.

I os c gbiogcejb suesbt dfL. Cppjyolm thb Fcl Whbdrbn yobjgs thctIhcs cffilotb sueecrI, scy I >{s1, . . . , sl}.

Whb ogbc fdr thb gbffilotodl dfE os thb fdjjdwolm;

Fdr 1 o l;Idlsogbr(so) clg gostolmuosh twd icsbs;

3. (so) 3 jm(so).Nc`b surb so((so) 3) os ol E

.

5. (so) 3 < jm(so).

Nc`b surb cjj ffilotb sbqublibs df jblmth(so)3 thct stcrt wothso clg crb ol crb ol E

.

Wd ciidnpjosh thos wb gbffilb E L ey;

s E o l P so((so) 3) >s ^

o l P so s jm(s) >(so) 3 (s) > 1 ^.

Ldtb thctE

os c gbiogcejb suesbt dfL

.]b wojj prdvb thct E os c ffilotb sueecr dfE .

E EJbts E .Ot fdjjdws frdn thb gbffilotodl dfE thct thbrb crb twd pdssoeojotobs;

3. o l P so((so) 3) >s ^.

Ey gbffilotodl df thos yobjgs s E .

5. o l P so s jm(s) >(so) 3 (s) > 1^.

Gbffilb L ey

(l) >g

s(l) ofl : jm(s),

>g

o P (l o) > 1 ^ of l jm(s).

, sd

(()) E. (5.6)

Cs so s > (jm(s)) clg (so)> 1,

() >(so) 3. (5.=)

51


21/94

Idneololm (5.6) clg (5.=) wb idlijugb,

((so) 3) E.

Cs (so) 3 >jm(s),

s> (jm(s)) >((so) 3) E.

Yd E E .

E os c ecr ol .Jbt .Gbtbrnolb o L suih thct(jm(so)) >so (I os c ecr ol ).Whbl ((so) 3) E

.

E os ffilotb.Fdr bvbry 1 o l thbrb crb twd pdssoeojotobs;

3. (so) 3 jm(so)so icusbs thb cggotodl df dlb bxtrc bjbnblt td E

.

5. (so) 3 < jm(so) os c fcl, sd fdr bcih t L woth (t) > 1 thbrb crb dljy ffilotbjyncly lcturcj lunebrs ` suih thct (t `) > 1.Yd thbrb crb dljy ffilotbjy ncly t df jblmth (so) 3 thct stcrtwoth so.

Cly bjbnblt ol E os ol thbrb ebicusb df bothbr rbcsdl 3 dr 5. Yd E dljyhcs ffilotbjy ncly bjbnblts.

HblibE os c ffilotb sueecr dfE , cs rbquorbg.

5.6.9 Bxtblgbg Fcl Whbdrbn

]b dftbl usb cl bxtblsodl df thb Fcl Whbdrbn whoih stctbs; of, fdr bvbry bjbnbltol c ibrtcol fcl, wb icl ffilg c lcturcj lunebr woth c gbsorbg prdpbrty, thbl wbicl ffilg L L suih thct, fdr bcih bjbnblt df thb fcl, wb icl ffilg c lcturcj

lunebr woth thb gbsorbg prdpbrty ebjdw L.

Gbffilotodl 5.55. Bxtblgbg Fcl WhbdrbnJbt eb c fcl clg U L suih thct fdr bcih thbrb bxosts l LsuihthctU(, l). Whbl wb icl ffilg L L suih thct l LP U(, l) ^.

Whbdrbn 5.59. CssunolmMCI3,1, thb Bxtblgbg Fcl Whbdrbn icl eb gbrovbgfrdn thb Fcl Whbdrbn.

Qrddf. Jbt eb c fcl clg U L woth thb prdpbrty

l L P U(, l) ^.

53


22/94

EyMCI3,1, thbrb bxosts c fulitodl ; L sctosfyolm

P U(, ()) ^.

Gbffilb c suesbt E dfL ey, fdr cjj s L,

s E (s)> 1.

E os gbiogcejb clg, cs os c fulitodl dl , E os c ecr ol . Yd wb icl cppjythb Fcl Whbdrbn clg ffilg N L suih thct

l NP (l)> 1 ^.

Gbffilb L>g

ncx( { (s) | jm(s) N} ). Whbl

l LP U(, l) ^,

cs rbquorbg.

Whb Bxtblgbg Fcl Whbdrbn os nuih strdlmbr thcl thb Fcl Whbdrbn clg, olidltrcst td thb Rlrbstroitbg Fcl Whbdrbn, ot gdbs ldt hdjg ijcssoicjjy. Dlbidujg gbffilb fdr bxcnpjb U I L ey, fdr cjj I, fdr cjj l L,

U(, l) (> 1 l> 1) (l)> 1.

Whb ijcssoicj ncthbnctoiocl wdujg scy thos rbjctodl sctosffibs;

I l L P U(, l) ^. (5.8)

Wd hon, cly bjbnblt dfIos bothbr bqucj td 1 (ol whoih icsb U(, 1)) dr os ldtbqucj td 1 (ol thct icsb hb os idlvolibg thbrb bxosts c pdsotodll woth(l)> 1).Ijbcrjy thbrb os ld lcturcj lunebrL, suih thct fdr cjjol Iwb icl ffilgl Lwoth U(, l). Hblib thb Bxtblgbg Fcl Whbdrbn gdbs ldt hdjg ijcssoicjjy.Whb oltuotodlostoi ncthbnctoiocl gdbs ldt cmrbb woth (5.8), hdwbvbr. Hblib thbrbjctodl U gdbs ldt sctosfy thb rbquorbnblts df thb Bxtblgbg Fcl Whbdrbn.

5.= Whb sbt df thb rbcj lunebrs

Ytcrtolm frdn thb twd ecsoi struiturbs wb hcvb oltrdguibg, L clg L, wb wclttd idlstruit ndrb idnpjbx struiturbs. Wd ebmol woth, wb gbffilb thb oltbmbrs ey[ >

g

(L L)/1, whbrb1 os thb bquovcjblib rbjctodl;

n, l 1 p, q n + q> l +p.

Cggotodl, nujtopjoictodl clg drgbr dl [ crb gbffilbg cs usucj.Whb rctodlcj lunebrs crb gbffilbg ey S >

g

([ L l p.

55


23/94

Cggotodl, nujtopjoictodl clg drgbr dl S crb gbffilbg cs usucj.Ldtb thct thb bqucjoty dl edth [ clg S os gbiogcejb.

Wd gbffilb thb rbcj lunebrs wb ffirst oltrdguib thb sbtYdf cjj (idgb lunebrs df)rctodlcj oltbrvcjs. Jbt eb c eokbitovb ncp frdn L tdS. ]b scy n L idgbsc rctodlcj oltbrvcj of clg dljy of thbrb bxost n1, n3 L suih thct

3. n> n1, n3,

5. (n1) (n3).

Fdr bcih n> n1, n3 ol Ywb gbffilb

n >g

(n1) clg n >g

(n3),

jm(n) >g n n.

Yd n clg n crb thb jbft clg thb romht blgpdolt df thb oltbrvcj idgbg ey nclg jm(n) os thb jblmth df thb oltbrvcj idgbg ey n.Fdr rctodlcj lunebrsp, q, wb wojj usucjjy kust wrotb Pp, q fdr thb rctodlcj oltbr-vcj3 (p), 3 (q).

Fdr cjj c, e Ywb gbffilb

c e >g

e c c e (c os idltcolbg ol e),

c e >g

c e e c (c tduihbs e).

Ldtoib wb cjsd gbffilbg c fulitodl jm clg c rbjctodl dl thb idjjbitodl df ffilotbsbqublibs df lcturcj lunebrs. Ey stctolm whoih idjjbitodl wb crb wdròlm wothot wojj eb ijbcr whoih ldtodl wb nbcl.

Gbffilotodl 5.57. Jbtx L.]b scy x os c rbcj lunebr, o.b. x U, of clg dljy of x os c shrolòlm clgshrovbjolm sbqublib df rctodlcj oltbrvcjs. Whct os,

(o). fdr cjj l L, x(l) Y,

(oo). fdr cjj l L, x(l+ 3) x(l) (shrolòlm)

(ooo). fdr cjj l L, thbrb bxosts n ol L suih thct jm(x(l)) 5n (shrovbjolm)

Fdr cjj x, y U, wb gbffilb

x y >g

l L P x(l) y(l) ^,

x#y >g

l L P ( x(l) y(l)) ^,

x : y >g

l L P x(l) : y(l), ^,

x y >g

l L P x(l) y(l) ^.

59


24/94

Ldtb thct bqucjoty dl thb rbcjs os ldt gbiogcejb. ]b idujg, fdr bxcnpjb, gbffilbc rbcj lunebr ey

(l) >g

P 3l , 3l

^ ofl `??,

>g

P 3` , 3` ^ of` > `?? clg ` l.

Cs dlb icl shdw

1 `??bxosts,

# 1 `??gdbs ldt bxost,

wb iclldt gbiogb 1 # 1.

Fcr ndrb icl eb scog cedut thb idlstruitodl df thb rbcj lunebrs, eut cs thos osldt df pcrtoiujcr onpdrtclib fdr thos thbsos wb gd ldt md oltd thos suekbit clyfurthbr.

57


25/94

Ihcptbr 9

Gosirbtb sprbcgs clg fcls

Ol thb prbvodus ihcptbr wb hcvb sbbl thct bqucjoty dl L os ldt gbiogcejb.Dlb idujg, hdwbvbr, stugy suesbts dfLthct gd hcvb c gbiogcejb bqucjoty, o.b.suesbtsC dfL woth

, C P > # ^. (9.3)

C suesbtC dfLsctosfyolm (9.3) os icjjbg gosirbtb.

Ol thos ihcptbr wb stugy gosirbtb sprbcgs clg fcls. ]b wojj sbb thct thbsbicl eb ihcrcitbrozbg ol c surprosolm clg loib wcy. Whb ihcrcitbrozctodl df gos-irbtb fcls jbcgs us td cl bquovcjblt df thb Fcl Whbdrbn, whoih wb prdvb ol thbjcst sbitodl df thos ihcptbr.

9.3 Gosirbtb sprbcgs

Ol sbitodl 5.9 wb hcvb oltrdguibg sprbcgs. ]hbl wdr`olm woth sprbcgs wb hcvbc pdwbrfuj tddj, thb (Mblbrcjozbg) Idltoluoty Qroliopjb (Gbffilotodl 5.31). RsolmMIQ wb wojj shdw thct c sprbcg os gosirbtb of clg dljy of ot os blunbrcejb.

Gbffilotodl 9.3. JbtC L.Cos blunbrcejbofi thbrb bxost 1, 3, . . . olL suih thctC>{1, 3, . . .}.

]b stcrt woth c jbnnc clg shdw; fdr cjj clg ol c gosirbtb sprbcg, wb icl

cjrbcgy gbiogb whbthbr > dr #cftbr `ldwolm ffilotbjy ncly vcjubs df clg .

Jbnnc 9.5. Jbt eb c gosirbtb sprbcg. Whbl

n L P n> n > ^.

Qrddf. Jbt eb c gosirbtb sprbcg.]b gbffilb fdr bcih L

dgg>g

((3), (9), (2), . . .) bvbl >g

((1), (5), (7), . . .).

52


26/94

Yonojcrjy, sdgg clg sbvbl crb gbffilbg fdr s L.

Gbffilb c sprbcg jcw ey, fdr cjj s L

,

(s) > 1 (sdgg) >(sbvbl) > 1.

Cs, fdr cjj , edth dgg clg bvbl crb ol thb gosirbtb sprbcg ,

P dgg> bvbl dgg# bvbl^. (9.5)

Gbffilb c rbjctodl U L ey

U 1 dgg> bvbl, U 3 dgg# bvbl.

Whbl, ey (9.5), fdr cjj ol , U1 dr U3. Yd usolm MIQ,

l, n L P n > n U l ^. (9.9)

Jbt .Gbffilb suih thct

dgg>g

clg bvbl >g

.

Whbl; U 1 clg ( U3) (cs dgg> > bvbl).Gbtbrnolbn L suih thct P n > n U1 ^ (sbb (9.9)). Whbl

P n > n dgg> bvbl^. (9.7)

Ldw jbt woth n> n.Gbffilb ey

dgg>g

clg bvbl >g

.

Whbl n> n, sd ey (9.7),

> dgg > bvbl > .

Clg wb idlijugb,

P n> n > ^.

]hct gdbs thos jbnnc tbjj us4 Jbt us vobw c sprbcg cs c trbb cmcol. Whbl thbjbnnc stctbs; of ydu crb wcj`olm up c trbb rbprbsbltolm c gosirbtb sprbcg, frdnsdnb pdolt dlwcrgs thbrb os dljy dlb wcy up (ol bcih ldgb thbrb os dljy dlbgorbitodl td ihddsb).Rsolm thos jbnnc, wb prdvb thct bvbry gosirbtb sprbcg os blunbrcejb. Wd gbffilbsuih cl blunbrctodl wb ffirst blunbrctb thb sbt df cjj ffilotb sbqublibs df lcturcjlunebrs. Fdr bcih cgnottbg ffilotb sbqublib (o.b. bcih ldgb ol thb trbb), wb

idlstruit cl bjbnblt df thb sprbcg ey stcrtolm ct thb movbl ffilotb sbqublib

56


27/94

clg ihddsolm, ct bcih stbp, thb jbftndst erclih. ]b cgg thb rbsujtolm olffilotbsbqublib td dur jost.

Fdr bvbry bjbnblt df thb gosirbtb sprbcg, wb icl ffilg c lcturcj lunebr nsuih thct cly sbqublib ol thb sprbcg stcrtolm woth n os bqucj td . Hblib,of ydu stcrt ol thb ldgb n clg tc`b, ol bcih stbp, thb jbftndst erclih up,ydu idlstruit (cs os thb dljy pcth ol thb sprbcg stcrtolm woth n), sd os ol dur blunbrctodl. ]b nc`b thos ogbc prbiosb ol thb prddf df thb fdjjdwolmthbdrbn.

Whbdrbn 9.9. Fdr bvbry sprbcg , os gosirbtb of clg dljy of os blunbrcejb.

Qrddf. Jbt eb c sprbcg.) Yuppdsb os gosirbtb.Jbts1, s3, . . . eb cl blunbrctodl dfL

.

Forst gbffilb ey, fdr cjj l L,

(l) >g

` P (l `) > 1 ^.

os wbjj-gbffilbg, cs os c sprbcg clg .]b gbffilb c sbqublib 1, 3, . . . cs fdjjdws;

Jbtp L clg idlsogbr sp.]b gostolmuosh twd icsbs;

3. (sp) > 3.Whbl sp os ldt cgnottbg ey .

Gbffilb p >g .5. (sp) > 1.

Gbffilb p ey;

p(l) >g

sp(l) ofl : jm(sp),

>g

` P (pl `) > 1 ^ of l jm(sp).

Ldtb thct thos sbqublib os wbjj-gbffilbg clg thct fdr bcih p, p .Yd{1, 3, . . .} .

Jbt .Gbtbrnolb, usolm Jbnnc 9.5, n L suih thct

P n > n >^. (9.2)

Gbtbrnolbp L suih thct n> sp.Ot fdjjdws frdn thb gbffilotodl dfp thct

pn> sp > n.

Cs p , ey (9.2), > p {1, 3, . . .}.Yd {1, 3, . . .}.

5=


28/94

Hblib > {1, 3, . . .} clg thbrbfdrb, os blunbrcejb.

) Yuppdsb os blunbrcejb.Jbt{1, 3, . . .} eb cl blunbrctodl df .Whbl,

l L P > l^.

Cppjyolm MIQ yobjgs;

n, l L P n> n > l^.

Jbt, .Gbtbrnolbn, l suih thct

P n> n > l^.

Ol pcrtoiujcr, cs n> n, > l. Hblib

P n> n > ^.

Whos nbcls thct, ey idlsogbrolm n clg n, wb crb cejb td gbiogb

> # .

Yd os gosirbtb.

9.5 Gosirbtb fcls

Bvbry fcl os ol pcrtoiujcr c sprbcg, sd Whbdrbn 9.9 cjsd hdjgs fdr gosirbtb fcls.Hdwbvbr, usolm thb Fcl Whbdrbn, wb icl prdvb cl bvbl strdlmbr stctbnblt fdrfcls, lcnbjy

bvbry gosirbtb fcl os c ffilotb sbt. (9.6)

Ol thb idltbxt df bjbnbltcry oltuotodlostoi clcjysos, thos stctbnblt os bquovcjblttd thb Fcl Whbdrbn.

Whbdrbn 9.7. Whb stctbnblt bvbry gosirbtb fcl os c ffilotb sbt os bquovcjblttd thb Fcl Whbdrbn

Qrddf.) Cssunb; bvbry gosirbtb fcl os c ffilotb sbt.Jbt eb c fcl clg E L c gbiogcejb ecr ol .]b gbffilb c fulitodl ; L {1, 3} ey;

(s) > 1 ( s l jm(s) P sl / E ^ )

( s l jm(s) P sl E n : l P sn /E ^

o L P l : o : jm(s) s(o) >k P (so k) > 1 ^ ^ ^ ).

58


29/94

Whb ogbc df os thct ol thb ebmollolm, ydu put cjj bjbnblts df cjsd ol , eut

dlib ydu crb ol E , ydu dljy tc`b thb jbftndst wcy up ol thb fcl .

os c fcl jcw, cs;

J3) Whb stctbnblt dl thb romht-hclg sogb df thb - soml os gbiogcejb, sd osc wbjj-gbffilbg fulitodl frdnL td{1, 3}. Ijbcrjy ( ) > 1.

J5) Fdr clys L woth(s) > 1,

(s k P( s k) > 1^ ) > 1.

Yd

s L P (s) > 1 l L P (s l) > 1^.

Whb dthbr gorbitodl () fdjjdws onnbgoctbjy frdn thb wcy os gbffilbg.

J9) Frdn thb fcit thct clg os c fcl, ot fdjjdws thct

s L P (s) > 1 ` L l L P (s l) > 1 l ` ^.

Furthbrndrb, os c gosirbtb fcl. ]b prdvb thos cs fdjjdws.

Jbt, .Gbtbrnolb ` L suih thct ` E .Ot fdjjdws frdn thb gbffilotodl df thct;

P ` > ` > ^.

Hblib, ey idlsogbrolm `, wb icl gbiogb;

> # .

Cs os c gosirbtb fcl, wb icl cppjy thb cssunptodl clg idlijugb thct osffilotb. Ycy >{1, . . . , p}.

Gbffilb, fdr 1 o l,

lo >g

k P okE ^.

Clg gbffilb;

E >g

{1l1, . . . , 1lp}.

Whbl ijbcrjy E E .

5?


30/94

Whb dljy tholm jbft td shdw os thct E os c ecr ol .

Jbt .

E os c gbiogcejb ecr ol , sd wb icl gbffilb l >g

o P o E ^.

Gbffilb L ey;

(`) >g

(`) of` l,

>g

k P ` k ^ of` < l.

Whbl , sd wb icl gbtbrnolb o L suih thct > o.Cs ol> l > l, ot fdjjdws frdn thb gbffilotodl dfl thct l > l o.Clg wb idlijugb

l> ol> olo E

.

Yd, fdr cjj ol , thbrb bxosts c lcturcj lunebr l suih thctl os ol E . HblibE os c ffilotb sueecr dfE.

) Cssunb thb Fcl Whbdrbn.Jbt eb c gosirbtb fcl.Ciidrgolm td Jbnnc 9.5,

n L P n> n > ^.

Rsolm thb Rlrbstroitbg Fcl Whbdrbn wb icl ffilg N Lsuih thct n N P n> n > ^.

Yd

, P N>N > ^. (9.=)

Cs os c fcl, thbrb crb dljy ffilotbjy ncly s L wothjm(s) >N clg(s) > 1.Idneololm thos woth (9.=) movbs thct os ffilotb, cs rbquorbg.

91


31/94

Ihcptbr 7

Dl dpbl, ijdsbg clg

blunbrcejb suesbts dfL

Ol Ihcptbr 2 wb wojj bxcnolb whbthbr thbrb bxost ihdoib fulitodls dl vcrodusidjjbitodls df suesbts dfL. Whb idjjbitodl df thb ijdsbg suesbts dfL clg thbidjjbitodl df thb dpbl suesbts df L crb twd df thbsb idjjbitodls. Ijcssoicjjy,thbrb crb c lunebr df bquovcjblt wcys td gbffilb thb ldtodls df ijdsbg clgdpbl. Oltuotodlostoicjjy, ldt cjj thdsb gbffilotodls crb bquovcjblt. Ebfdrb wb iclmd oltd thb prdejbn df thb bxostblib df ihdoib fulitodls wb hcvb td eb ijbcr dlwhct idjjbitodls wb crb tcj`olm cedut. Ol thos ihcptbr wb stugy thb gofibrbltgbffilotodls clg thbor gbpblgbliobs.

7.3 Cjj `olgs df ijdsbg clg dpbl

Ol ijcssoicj ncthbnctois thb ndst idnndljy usbg gbffilotodl df dpbl os thb dlbthct stctbs; c suesbtC dfLos dpbl ofi bvbry bjbnblt dfC hcs c lbomhedurhddgthct os idltcolbg ol C. IdlsogbrolmLwoth thb nbtroi movbl ey

g(, ) >g

l>1

sm((l) (l)) 5l,

whbrb, fdr bvbry lcturcj lunebr n,

sm(n) > 1 n> 1 clg sm(n) > 3 n> 1,

thos ldtodl df dpbl idrrbspdlgs woth thb oltuotodlostoi ldtodl df wbc`jy dpbl.Dlb idujg cjsd gbffilb cl dpbl suesbt dfL cs c ulodl df ecsoi dpbl suesbts dfL, whbrb c suesbtC dfL os ecsoi dpbl ofi thbrb bxosts s L woth

C> { L | (jm(s)) >s }.

Whos gbffilotodl idrrbspdlgs woth thb oltuotodlostoi ldtodl df bfibitovbjy dpbl,prdvogbg wb stcrt frdn c gbiogcejb idjjbitodl df ecsoi dpbl sbts, o.b. C L osbfibitovbjy dpbl ofi thbrb bxosts c gbiogcejb suesbt E dfL suih thct

C>sE

{ L | (jm(s)) >s}.

93


32/94

C prbiosb gbffilotodl df thbsb twd ldtodls df dpbl os thb fdjjdwolm.

Gbffilotodl 7.3. JbtM L.

]b scy M os wbc`jy dpblof clg dljy of fdr bvbry M, thbrb bxost l L, suihthct bvbry L pcssolm thrdumh l ebjdlms td M.]b scy M os bfibitovbjy dpbl of clg dljy of thbrb bxosts Lsuih thct, fdr cjj L, ebjdlms td M of clg dljy of thbrb bxosts l L woth l G (o.b.(l) > 3).

Ol thb gbffilotodl df bfibitovbjy dpbl wb idujg kust cs wbjj rbpjcib G eyB. Wdsbb thos, gbffilb fdr bcih L cl bjbnblt dfL ey;

(l) >g

3 of` l P (`)> 1 (`) 3 l ^,

>g

1 dthbrwosb.

Whbl;

{ L |l P l B^} > { L |l, ` P (`) >l + 3 ^},

> { L |n P (n) > 3 ^},

> { L |n P n G^}.

Dlb icl bcsojy shdw thct bvbry bfibitovbjy dpbl suesbt dfLos cjsd wbc`jy dpbl.Whb idlvbrsb os ldt trub oltuotodlostoicjjy. ]b wojj shdw thos usolm thb YbidlgCxodn df Idltoludus Ihdoib (MCI3,3).

Whbdrbn 7.5. Ldt; bvbry wbc`jy dpbl suesbt dfL os bfibitovbjy dpbl.

Qrddf. Yuppdsb bvbry wbc`jy dpbl suesbt dfL os bfibitovbjy dpbl.Gbffilb, fdr bvbry L, c suesbt M dfL ey

M>g

{ L | l P (l) > 1 ^}.

M os wbc`jy dpbl fdr bcih L (kust poi` l > 3 fdr bvbry M). Yd frdnthb cssunptodl ot fdjjdws thct, fdr bcih ol L, M os bfibitovbjy dpbl, o.b.;

L L LP M l P (l) > 3 ^ ^.

Ciidrgolm tdMCI3,3, thbrb bxosts c fulitodl ; L L suih thct

L LP M l P (|)(l) > 3 ^ ^.

Idlsogbr > 1.M1 > L, sd ol pcrtoiujcr 3 M1. Hblib,

l P (|1)(3l) > 3 ^.

Gbtbrnolbl suih thct (|1)(3l) > 3 clg gbffilb n >g

3l.

Gbtbrnolb` suih thctn(1`)> 1.

95


33/94

Ey gbffilotodl, (|1)(n) >n(1).Yd, fdr cjj L woth ` > 1`, n() >n(1) > 3. Hblib,

LP `> 1` 3 M^. (7.3)

Gbffilb L ey;

(l) >g

1 ofl `,

>g

3 ofl < `.

Whbl`> 1` clg ey (7.3), 3 M.Eut(`+ 3) > 3, sd l P (l) > 1 ^ clg M>. Idltrcgoitodl.]b idlijugb; ldt bvbry wbc`jy dpbl suesbt dfLos bfibitovbjy dpbl.

Jbt us ndvb dl td thb ijdsbg suesbts dfL. Wd ebmol woth, wb movb c gbffilotodldf thb ldtodl df thb idnpjbnblt df c suesbt dfL.

Gbffilotodl 7.9. JbtC L.Whb idnpjbnblt dfC, ldtctodl C, os gbffilbg ey

C >g

{ L | /C}.

Yd C os thb sbt df cjj olLfdr whoih thb cssunptodl ebjdlms td C jbcgstd c idltrcgoitodl. Ldtb thct ldt fdr bvbry suesbt C dfL, (C) >C.

Hcvolm twd gofibrblt ldtodls df dpbl, thb ogbc; c sbt os ijdsbg of clg dljy of ot osthb idnpjbnblt df cl dpbl sbt, cjsd jbcgs td twd ldtodls df ijdsbg suesbt dfL.Ldtb thct wb scy ot os thb idnpjbnblt df cl dpbl sbt clg ldt ots idnpjbnbltos dpbl. Ijcssoicjjy thbsb twd stctbnblt crb bquovcjblt, eut oltuotodlostoicjjythby crb ldt.]b cjsd oltrdguib c thorg ldtodl df ijdsbg, lcnbjy sbqubltocjjy ijdsbg.

Gbffilotodl 7.7. JbtF L.Whb sbtF oswbc`jy ijdsbgof clg dljy of thbrb bxosts c wbc`jy dpbl suesbt M dfL suih thct Fos thb idnpjbnblt dfM.]b scyF osbfibitovbjy ijdsbgof clg dljy ofFos thb idnpjbnblt df cl bfibitovbjy

dpbl sbt. Dr, bquovcjbltjy, Fos bfibitovbjy ijdsbg of clg dljy of thbrb bxosts Lsuih thct, fdr cjj L, ebjdlms td F ofi fdr cjj l L, l G.]b icjjFsbqubltocjjy ijdsbgof clg dljy of, fdr bvbry L, of, fdr bvbryl L,thbrb bxosts F pcssolm thrdumh l, thbl otsbjf ebjdlms td F.

]b wojj ldw olvbstomctb hdw thdsb gofibrblt ldtodls df ebolm ijdsbg crb rbjctbg.

99


34/94

Whbdrbn 7.2. Bvbry wbc`jy ijdsbg suesbt dfL os sbqubltocjjy ijdsbg.

Qrddf. JbtFeb c wbc`jy ijdsbg suesbt dfL.

Yuppdsb L clg l L FP l > l ^. ]b hcvb td prdvb; F.Cs F os wbc`jy ijdsbg, wb icl ffilg c wbc`jy dpbl sbt M suih thct F > M.Ldtb thct;

F /M,

( lP l > l M ^ ),

l ( P l > l M ^ ).

Jbtl L.Ey cssunptodl, thbrb bxosts F, suih thct l> l.Cs F, ( P l> l M ^ ). Hblib, usolm l> l,

( P l > l M ^ ).

Whos hdjgs fdr cjj l L, sdF clg F os sbqubltocjjy ijdsbg.

Whb idlvbrsb os ldt trub idlstruitovbjy; ldt bvbry sbqubltocjjy ijdsbg suesbt oswbc`jy ijdsbg. Jbt Qeb cl ulsdjvbg ncthbnctoicj prdejbn clg gbffilb

F >g

{ L | Q Q}.

F os sbqubltocjjy ijdsbg, eut ldt wbc`jy ijdsbg.Fdr suppdsb F os wbc`jy ijdsbg. Whbl wb icl ffilg c wbc`jy dpbl sbt M suih

thctF >M clg, fdr cjj L,

( F) ( /M).

Cs( /M) os bquovcjblt td /M, fdr cjj L,

( F) M >F.

Yd of(Q Q), thbl Q Q.Eut(Q Q) os trub oltuotodlostoicjjy clg Q Qncy eb rbi`jbss P2^.

Ot os ijbcr thct bvbry bfibitovbjy ijdsbg suesbt dfL os cjsd wbc`jy ijdsbg. Hdw-

bvbr, cs ol thb icsb df thb dpbl suesbts dfL, thb idlvbrsb gdbs ldt hdjg. Whbprddf df thos thbdrbn rbsbnejbs thb prddf df Whbdrbn 7.5.

Whbdrbn 7.6. Ldt; bvbry wbc`jy ijdsbg suesbt dfL os bfibitovbjy ijdsbg.

Qrddf. Kust cs ol thb prddf df Whbdrbn 7.5, gbffilb, fdr bvbry L,

M>g

{ L |l P (l) > 1 ^}.

M os wbc`jy dpbl, clg thbrbfdrb, F>g

(M) os wbc`jy ijdsbg.

YuppdsbF os bfibitovbjy ijdsbg fdr bcih , thbl;

P F l P (l) > 3^ ^.

97


35/94

EyMCI3,3, thbrb bxosts c fulitodl ; L L suih thct

P F l P (|)(l) > 3 ^ ^.

Idlsogbr 1.M1 > L, sdF1> clg ol pcrtoiujcr

3 /F1. (7.5)

Yuppdsbl P (|1)(3l)> 3 ^.

Folg suih l1 clg ffilg n suih thct

P 1n> n (|)(3l1)> 3 ^. (7.9)

Gbffilb L ey

(l) >g 1 ofl n,

>g

3 ofl < n.

Whbl n > 1n sd, ey (7.9), (|)(3l1)> 3 clg 3 /F.Eut(n+ 3) > 3, sdl P (l) > 1 ^ clgF >L. Idltrcgoitodl.

Hblibl P (|1)(3l)> 3 ^, whoih nbcls

l P (|1)(3l) > 3 ^.

Ey gbffilotodl df, 3 F1, idltrcgoitolm (7.5).]b idlijugb; ldt bvbry wbc`jy ijdsbg suesbt dfL os bfibitovbjy ijdsbg.

]b sunncrozb thb rbsujts dl ijdsbg suesbts dfL cs fdjjdws;Fdr bvbry suesbtC dfL,

C os bfibitovbjy ijdsbg C os wbc`jy ijdsbg C os sbqubltocjjy ijdsbg.

Ol Ihcptbr 5 wb hcvb oltrdguibg sprbcgs (Gbffilotodl 5.8). Bvbry sprbcg ossbqubltocjjy ijdsbg. Ol cggotodl, sprbcgs crb jdictbg suesbts dfL. C sbt C dfL os icjjbg jdictbg of clg dljy of wb icl gbiogb, fdr cjj s L, whbthbr thbrbbxosts ol C pcssolm thrdumh s dr ldt.

Bxcnpjb 7.=. Idlsogbr thb suesbt C dfL gbffilbg ey;

C > 1 ( > 3 `??gdbs ldt bxost ).

Cos cl bfibitovbjy ijdsbg suesbt dfL. Fdr wb icl gbffilb L ey;

(l) > 3 n P l> 1n (l> 3n n : `??) ^.

Whbl,

C l P (l) > 3 ^.

Yd ol pcrtoiujcr C os sbqubltocjjy ijdsbg. Hdwbvbr, C os ldt c sprbcg, ebicusbC os ldt jdictbg; wb iclldt gbiogb whbthbr dr ldt thbrb bxosts ol C pcssolm

thrdumh 3 .

92


36/94

7.5 Ybqubltocjjy ijdsbg suesbts dfL

Ol thb prbvodus sbitodl wb hcvb oltrdguibg thrbb ldtodls df ijdsbg. Ol thossbitodl wb stugy dlb df thbn, lcnbjy thb ldtodl; sbqubltocjjy ijdsbg. Durcon os td ffilg c wcy td ihcrcitbrozb ffilotbjy blunbrcejb clg blunbrcejb suesbtsdfL thct crb sbqubltocjjy ijdsbg.

Gbffilotodl 7.8. JbtC L.]b gbffilb thb ijdsurb df C, ldtctodlC, ey

C >g

{ L |l L C P l >l} ^.

Ldtb thct c suesbt C dfLos sbqubltocjjy ijdsbg of clg dljy ofC > C.

Furthbrndrb, fdr cly suesbt C dfL, C os sbqubltocjjy ijdsbg. Fdr jbtC L.Ot os ijbcr thct C C. Ldw suppdsb C.Wd prdvb C, wb hcvb td ffilg, fdr cjj l L, C woth l > l.

Jbtl L.Cs C, thbrb bxosts C, woth

l > l. (7.7)

Cs C, thbrb bxosts C, woth

l > l. (7.2)

Idneololm (7.7) clg (7.2) yobjgs,

l > l.

Hblib, fdr cjj l L, thbrb bxosts C woth l> l, clg thbrbfdrb, C.

Yd C > C.

7.5.3 Folotbjy blunbrcejb suesbts dfL

Forst wb idlsogbr thb ffilotbjy blunbrcejb suesbts df L. C suesbt C df L os

icjjbg ffilotbjy blunbrcejb ofiC os df thb fdrn {1, . . . , `}, fdr ibrtcol ` Lclg o L. Ldt bvbry ffilotbjy blunbrcejb suesbt dfL os sbqubltocjjy ijdsbg.Gbffilb fdr bxcnpjb 1, 3 L ey;

1>g

1.

3(l) >g

1 ofl `??,

>g

3 ofl < `??.

96


37/94

Ijbcrjy{1, 3} os ffilotbjy blunbrcejb. Eut thos sbt os ldt sbqubltocjjy ijdsbg.Fdr wb ncy gbffilb L ey

(l) >g

1 ofl `?? (l < `?? `?? os dgg),

>g

3 ofl < `?? `??os bvbl.

Whbl, fdr cjj l L, l> 1l l > 3l, sd {1, 3}.Ldtb thct;

> 1 l L P l> `?? los bvbl ^,

> 3 l L P l> `?? los dgg^.

Whbrbfdrb, wb iclldt gbiogb > 1 > 3. Hblib wb hcvb ld rbcsdl td

crn {1, 3}clg{1, 3} os ldt sbqubltocjjy ijdsbg.

Wd ihcrcitbrozb thb ffilotbjy blunbrcejb sbqubltocjjy ijdsbg suesbts dfLwb ffirstshdw thct thbsb crb sprbcgs. Whos cjjdws us td usb thb Idltoluoty Qroliopjb tdprdvb cly furthbr prdpbrtobs df thbsb suesbts dfL.

Jbnnc 7.?. Fdr cjj` L, fdr cjj1, . . . , ` L,

{1, . . . , `} os c sprbcg.

Qrddf. Jbt` L clg 1, . . . , ` L.Gbffilb ; L {1, 3} ey, fdr cjj s L,

(s) > 1 o ` P s> o(jm(s)) ^.

Whb stctbnblt dl thb romht-hclg sogb os gbiogcejb sd os c wbjj-gbffilbg fulitodl.Ot os ijbcr thct os c sprbcg jcw (gbffiltodl 5.8).Ldtb thct;

l P (l) > 1 ^,

l oP l> ol ^,

{1, . . . , `}.

Yd > {1, . . . , `} clg{1, . . . , `} os c sprbcg.

C suesbt C dfL os icjjbg ffilotb ofi thbrb bxost ` L clg 1, . . . , ` L suihthct C> {1, . . . , `} clg ol cggotodl, fdr cjj o : k `, o # k . Ol thb lbxtthbdrbn wb prdvb thct c ffilotbjy blunbrcejb suesbt dfL os sbqubltocjjy ijdsbgof clg dljy of ot os ffilotb.

9=


38/94

Whbdrbn 7.31. Fdr cjj` L, fdr cjj1, . . . , ` L,

{1, . . . , `}> {1, . . . , `} o, k ` P o > k o # k^.

Qrddf. Jbt` L.) Yuppdsbo, k ` P o > k o # k^.

Ot os ijbcr thct {1, . . . , `} {1, . . . , `}.

Jbt {1, . . . , `}.]b icl cssunb, fdr cjj o : k `, o # k .Gbffilb

N >g

ncx( { p P op>kp ^ | o : k ` } ).

Whblo, k` P o>k oN>kN^.

Cs {1, . . . , `}, ey gbffilotodl

lo ` P l > ol ^.

Folg o suih thct N>oN.Whbl> o {1, . . . , `}.Yd{1, . . . , `} {1, . . . , `}.

) Yuppdsb{1

, . . . , `}> {1

, . . . , `}.Whbl

{1, . . . , `} o ` P > o^.

Cs {1, . . . , `} os c sprbcg (Jbnnc 7.?), wb icl cppjy thb Idltoluoty Qroliopjb.Whos yobjgs

{1, . . . , `} l, o {1, . . . , `} P l >l > o^. (7.6)

Ldw jbt o, k `.]b hcvb td shdw; o > k o # k .

Cs o {1, . . . , `}, wb icl ffilg, ey cppjyolm (7.6), l L suih thct

{1, . . . , `} P ol> l > o^.

]b gostolmuosh twd icsbs;

3. ol>kl.Whbl o# k .

5. ol> kl.

Ldtb thct k {1, . . . , `}, clg thbrbfdrb, o > k .

Yd wb icl gbiogb; o> k o # k , cs rbquorbg.

98


39/94

7.5.5 Blunbrcejb suesbts dfL

Ol thb prbvodus sbitodls wb hcvb fdulg c wcy td ihcrcitbrozb ffilotbjy blunbrcejb

suesbts C dfL woth thb prdpbrty C> C. Whos nc bs us wdlgbr whbthbr wbicl ffilg suih c ihcrcitbrozctodl fdr blunbrcejb suesbts C dfL cs wbjj. WhbIdltoluoty Qroliopjb pjcybg c iruiocj rdjb ol thb prddf df thb ffilotbjy blunbrcejbicsb. ]b wbrb cjjdwbg td usb thos proliopjb, ebicusb C os c sprbcg fdr bvbryffilotbjy blunbrcejb suesbt C df L. Rlfdrtulctbjy, thos os ldt ol mblbrcj thbicsb fdr blunbrcejb suesbts dfL.

Bxcnpjb 7.33. Idlsogbr thb sbt {1, 3, . . .}, gbffilbg ey

l>g

1 ofl : `??,

>g

3 ofl `??.

Yuppdsb{1, 3, . . .} os c sprbcg, scy .

Whbl wb crb cejb td gbiogb

(3) > 1 (3)> 1.

Ldtb thct

(3) > 1 `?? bxosts.

Hblib wb icl gbiogb

`?? bxosts `??gdbs ldt bxost.

Eut thos stctbnblt os rbi`jbss.

Whos os c Erduwbrocl idultbrbxcnpjb. Rsolm thb Idltoluoty Qroliopjb, dlbicl detcol c idltrcgoitodl frdn thb stctbnblt; fdr bvbry L, thb sbt{1, 3, . . .} os c sprbcg.Ol thb bxcnpjb cedvb {1, 3, . . .}> {1, 3, . . .}. Fdr suppdsb {1, 3, . . .}.Whbl wb icl gbtbrnolbo L suih thct3 >o3, whblib> o {1, 3, . . .}.

Gbffilotodl 7.35. JbtX L.X os icjjbg stroitjy clcjytoi ofi thbrb bxosts c idltoludus fulitodl f ; L Lsuih thct;

X> X>Ucl(f).

]b wojj sbb thct thb sbt {1, 3, . . .} os stroitjy clcjytoi fdr bcih L. Whoscjjdws us td usb thb Idltoluoty Qroliopjb lbvbrthbjbss whbl prdvolm prdpbrtobsdf thb blunbrcejb sbqubltocjjy ijdsbg suesbts dfL.

9?


40/94

Jbnnc 7.39. Jbt L clg gbffilbC>g

{1, 3, . . .}.

OfC os sbqubltocjjy ijdsbg, o.b. C> C, thbl, fdr cjj1 C, thbrb bxostsl L,

suih thct, fdr cjj3 C woth1l> 3l, 1> 3.

Qrddf. Forst wb shdw C os stroitjy clcjytoi ey gbffilolm c idltoludus fulitodlf ;L L wothC> Ucl(f). Ytcrt frdn thb desbrvctodl thct fdr bvbry L,

C l n P l > nl ^.

Hblib, ey cppjyolm thb Noloncj Cxodn df Ihdoib 3.

C L l P l > (l)l ^.

]b gbffilb thb fulitodl f cs fdjjdws;

f()(l) >g

(l)(l) of` l P f()`> (`)` ^,

>g

(n)(l) of` l P f()`>(`)` ^

clg n> o P f()(o + 3)>(o+3)(o + 3) ^.

]b ijcon; C> Ucl(f). Dlb prdvbs thos cs fdjjdws.

) Jbt C.Whbl, ey gbffilotodl,

l n P l > nl ^.

Ciidrgolm td thb Noloncj Cxodn df Ihdoib thbrb bxosts Lsuih thct

l P l > (l)l ^.

Ot fdjjdws frdn thb gbffilotodl dff thct> f() Ucl(f).

) Jbt Ucl(f).Folg suih thct > f().Dlb icl prdvb thct l n P f()l> nl ^.Hblib C.

Ot os ijbcr thct f os idltoludus cs

1, 3 LP 1l> 3l f(1)l> f(3)l ^. (7.=)

Ldw wb crb rbcgy td prdvb thb jbnnc.CssunbC > C clg jbt feb cs gbffilbg cedvb.Cs Ucl(f) >C > C,

L o L P f() >o ^.

3Whb Noloncj Cxodn df Ihdoib os CI1,1cs gbffilbg dl pcmb 32, eut woth thb bxtrc idlgotodl

thct Q LL eb gbiogcejb. Ol thct icsb, dlb ncy gbffilb L ey(l) >g n PQ(l,n) ^.

71


41/94

Cppjyolm IQ yobjgs

L n, o L LP n> n f() >o ^.

Jbt1 C clg gbtbrnolb L suih thct 1> f().Gbtbrnolb lcturcj lunebrs n clg osuih thct

LP n> n f() > o^.

Whbl ol pcrtoiujcr 1> f() >o, clg thbrbfdrb,

LP n> n f() > 1^. (7.8)

Jbt3 C woth 1n> 3n.]b hcvb td shdw; 1> 3

Gbtbrnolb L suih thct 3 > f() clg gbffilb L ey

(`) >g

(`) of : n,

>g

(`) of n.

n > n, sd usolm (7.=)

f()n> f()n> 1n> 3n> f()n.

Frdn pjcibn dlwcrgsclgcrb thb scnb. Yd ey thb wcy fos gbffilbg wb sbb

f() >f() >3. (7.?)

Dl thb dthbr hclg n> n, sd ey (7.8)

f() >f() >1. (7.31)

Idneololm (7.?) clg (7.31), wb idlijugb 1> 3.

Fdr bvbry suesbt C df L, cl bjbnblt df C os icjjbg cl osdjctbg pdolt df Cofi thbrb bxosts c lcturcj lunebr l suih thct cjj bjbnblts dfC thct stcrt wothl crb bqucj td . Jbnnc 7.39 shdws thct oltuotodlostoicjjy cjj bjbnblts df csbqubltocjjy ijdsbg suesbt df L crb osdjctbg. Whos os ldt thb icsb ijcssoicjjy.Idlsogbr fdr bxcnpjbC L gbffilbg ey

C>g { 1,3 1,1, 3 1,1, 1, 3 1, . . . }.

Frdn c ijcssoicj pdolt df vobw thos sbt os sbqubltocjjy ijdsbg, eut 1 os ldt osdjctbgol C cs thbrb os ld lcturcj lunebr l suih thct bvbry bjbnblt dfCthct stcrtswoth 1los bqucj td 1.Ldtb thct idlstruitovbjy C os ldt sbqubltocjjy ijdsbg. ]b idujg gbffilb Ley, fdr cjj l L,

(l) >g

1 ofl >`??,

>g

3 ofl > `??.

WhblC, eut ot os rbi`jbss td scy C.

73


42/94


43/94

Whbdrbn 7.3=. Fdr bvbry L,

{1, 3, . . .}> {1, 3, . . .} Whbrb bxosts1, s3, . . . L clg1, 3, . . . L suih thct

(o) {1, 3, . . .}> {s1 1, s3 3, . . .},

(oo) o, k L P o>k so sk^,

(ooo) {1, 3, . . .} o L P (jm(so)) >so ^.

Qrddf. Jbt Lclg gbffilb C >g

{1, 3, . . .}.

) CssunbC > C.]b gbffilbs1, s3, . . . L

clg1, 3, . . . Lstbp ey stbp. ]hojb gbffilolm thbsbsbqublibs wb cjsd gbffilb c thorg sbqublib n1, n3, . . . L, whoih wb usb jctbrdl td shdw thct thb ffirst twd sbqublibs hcvb thb rbquorbg prdpbrtobs.

Ytbp 3 Ciidrgolm td Jbnnc 7.39 wb icl gbtbrnolbn1 L suih thct

C P n1> 1n1 > 1 ^.

Gbffilb s1>g

1n1 clg 1>g

1 Yn1 .

Ytbp l+3 Yuppdsb wb hcvb cjrbcgy gbffilbgs1, . . . s` clg 1, . . . .Idlsogbrl+3.Ey Idrdjjcry 7.37, o P o >l+3 o #l+3 ^.Gostolmuosh twd icsbs;

3. o : l + 3 P o >l+3 ^.

Gbffilb nl+3>g 1 clg idltolub woth stbp l + 5.

5. o : l + 3 P o #l+3 ^.Gbtbrnolb nl+3 suih thct

C P nl+3> l+3nl+3 >l+3 ^.

Gbffilb s`+3>g

l+3nl+3 clg +3>g

l+3 Ynl+3.

Ldw wb hcvb td shdw s1, s3 . . . clg 1, 3 . . . hcvb prdpbrtobs (o), (oo) clg (ooo).

(o). {1, 3, . . .}> {s1 1, s3 3, . . .}.

) ]b wojj prdvb thos woth thb proliopjb df idursb-df-vcjubs-olguitodl.Jbtl L clg cssunb fdr cjj o : l, o {s1 1, s3 3, . . .}.

Icsb 3; o : l P o >l ^.Gbtbrnolb suih o.Cs o : l wb icl cppjy thb olguitodl hypdthbsos clg ffilg ` Lwoth o >s` .Yd l >o >s` {s1 1, s3 3, . . .}.

79


44/94

Icsb 5; o : l P o #l ^.Whos nbcls thct ol thb idlstruitodl df s1, s3 . . . clg 1, 3 . . .

lbw s` clg crb cggbg td thb jost ol stbp l.Gbtbrnolb` suih thct s`, crb gbffilbg ol stbp l.Whbl s` > lnl clg >

l Ynl .Hblibl > s` {s1 1, s3 3, . . .}.

) Jbt ` L.]b icl gbtbrnolb l L suih thct s` > lnl clg >

l Ynl .Yd s` >lnl (

l Ynl) >l {1, 3, . . .}.

(oo). o, k P o>k so sk^.Jbto, k L woth o >k . ]b wojj prdvb s o sk .Gbtbrnolb p, q L suih thct so os idlstruitbg ol stbp p clg sk os idl-

struitbg ol stbp q. Ot fdjjdws frdn thb wcy thbsb crb idlstruitbg thct

p #q . (7.33)

Yuppdsbso sk .

Whbl np nq clg thbrbfdrb, cs qnq >sk ,

qnp> so > pnp.

Ey gbffilotodl dfnp,

C P np > pnp > p ^.

Yd ol pcrtoiujcr, cs q Cclg qnp> so> pnp,

q >p.

Idltrcgoitolm (7.33).

Hblibsosk .Ol c sonojcr wcy wb sbb thct sk so clg wb idlijugb so sk .

(ooo). {1, 3, . . .} o P (jm(so)) >so ^.Jbt {1, 3, . . .}.

]b hcvb td ffilg o L woth (jm(so)) >so.Cs

{1, 3, . . .}>{1, 3, . . .}> {s1 1, s3 3, . . .},

wb sbb thct

{s1 1, s3 3, . . .}.

Gbtbrnolb o suih thct >so o.Whbl (jm(so)) >so.

77


45/94

) Yuppdsb thbrb bxost s1, s3, . . . L clg 1, 3, . . . L sctosfyolm rbquorb-

nblts (o), (oo) clg (ooo).

Ot os ijbcr thct C C.

Jbt C.Ey prdpbrty (ooo) wb icl gbtbrnolb o L woth (jm(so)) >so.Rsolm prdpbrty (oo) wb sbb thct, fdr cjj k L woth o >k,

(jm(so))>sk k(jm(so)). (7.35)

]b hcvb cssunbg C >{s1 1, s3 3, . . .}, clg thbrbfdrb,

l L k L P l > sk kl ^. (7.39)

Idneololm (7.35) clg (7.39) yobjgs

l L P l > so ol ^.

Yd > so o {s1 1, s3 3, . . .}>{1, 3, . . .}.

72


46/94

Ihcptbr 2

Dl thb bxostblib clg

ldl-bxostblib df ibrtcol ihdoib

fulitodls

Ol thos ihcptbr wb stugy thb fdjjdwolm vbrsodl df thb Cxodn df Ihdoib, fdrnu-jctbg ey [brnbjd ol 3?18;

fdr bvbry fcnojy Fdf ldl-bnpty sbts, thbrb bxosts c fulitodl f ;F

Fsuih thct fdr bcih Y F ;f(Y) Y.

Movbl c fcnojy F, c fulitodl sctosfyolm thb cedvb idlgotodl os icjjbg c ihdoib

fulitodl dl F. ]b wojj idlsogbr vcrodus fcnojobs df ldl-bnpty sbts clg ol-vbstomctb whbthbr dlb icl gbffilb, ol cl bfibitovb wcy, c ihdoib fulitodl dl thbfcnojy.

2.3 Ihdoib fulitodls dl idjjbitodls df suesbts dfL

Whb Cxodn df Ihdoib gbcjs woth idjjbitodls df ldl-bnpty sbts. Oltuotodlostoicjjyc suesbtC dfL os icjjbg ldt bnpty ofil P l /C ^. Ldtb thct thos os bquovcjblttdl P l C ^. ]b wclt td gbffilb c strdlmbr ldtodl.

Gbffilotodl 2.3. JbtC L.

Cos olhceotbg of clg dljy of thbrb bxosts c lcturcj lunebr l woth l C.C suesbtC dfLncy eb ldt bnpty whojb wb crb ulcejb td prdvb C os olhceotbg.Fdr bxcnpjb, jbtQeb c stctbnblt wb icl, ct thb ndnblt, ldt prdvb ldr rbkbit(fdr bxcnpjb thb Uobncll Hypdthbsos) clg gbffilb;

C>g

{l L | (l> 1 Q) (l> 3 Q)}.

Cs wb icl prdvb (Q Q), wb hcvb (1 C 3 C), whoih nbcls C osldt bnpty. Eut wb iclldt prdvb 1 C, ldr 3 C, sd C os ldt olhceotbg.

76


47/94

Ol thb spbiocj icsb df gbiogcejb suesbts dfL, fdr cly L, stctolm G osolhceotbg clg G os ldt bnpty nbcl, rbspbitovbjy,

l P (l) > 3 ^ clg l L P (l) > 3 ^.

Ncr`dvs Qroliopjb scys thbsb twd stctbnblts crb bqucjjy strdlm. Whos proliopjbos ciibptbg ol sdnb erclihbs df idlstruitovb ncthbnctois. Hdwbvbr, usolm thbErduwbr-@rop`b-cxodn dlb icl idlstruit c sbqublib (1), (3), . . . suih thctthb sbidlg stctbnblt os oltuotodlostoicjjy trub, eut thb ffirst dlb os rbi`jbss P7^.]b wojj dljy idlsogbr idjjbitodls df olhceotbg suesbts dfL. Ol thb rbncolgbr dfthb thbsos wb wojj usb edth thb tbrn ldl-bnpty clg thb tbrn olhceotbg fdrthb ldtodl gbffilbg ol 2.3.

Whbdrbn 2.5. GbffilbC>g

{ L | G os olhceotbg}.

Whbrb bxosts c fulitodlf ;C L suih thct

C P (f()) > 3 ^.

Qrddf. Jbt C clg gbffilb;

f() >g

o P (o) > 3 ^.

Cs C wb `ldw G os olhceotbg, sd f os wbjj-gbffilbg.Ijbcrjy(f()) > 3.

Cppcrbltjy, wb icl gbffilb c ihdoib fulitodl dl thb idjjbitodl df olhceotbggbiogcejb suesbts dfL. Ldw jbt us idlsogbr thb idjjbitodl df olhceotbgjy blu-nbrcejb suesbts dfL. Whbsb crb gbffilbg cs fdjjdws.

Gbffilotodl 2.9. Fdr cly Lwb gbffilb

OB>g

{(l) | l L}.

JbtX L.]b scy thct olhceotbgjy blunbrctbs X ofiX idoliogbs woth OB.Xos cl olhceotbgjy blunbrcejb suesbtdfL ofi sdnb bjbnblt dfL olhceotbgjyblunbrctbsX.

Wwd gofibrblt bjbnblts dfLncy olhceotbgjy blunbrctb thb scnb suesbt dfL.Idlsogbr, fdr bxcnpjb,>

g

(3, 5, 5, 5, 5, . . .) clg >g

(5, 3, 3, 3, 3, . . .).

OB> {3, 5}> O B,

eut >.]hbl gbffilolm c ihdoib fulitodl dl thb idjjbitodl df olhceotbgjy blunbrcejbsuesbts dfLusolm thos pcrcnbtrozctodl, wb df idursb wclt thb ihdoib fulitodltd cssoml thb scnb vcjub td thb cedvb gbffilbg clg . Clg ol mblbrcj, wbwclt, fdr cjj , ol L woth OB > OB, f() > f(). Rlfdrtulctbjy, thos os

onpdssoejb, cs thb fdjjdwolm thbdrbn pdolts dut.

7=


48/94

Whbdrbn 2.7. Whbrb os ld fulitodlf ;L L suih thct

(o). L l L P f() >(l) ^,

(oo). , LP OB > O B f() >f() ^.

Qrddf. Yuppdsbf ;L L sctosffibs thbsb rbquorbnblts.Cppjyolm thb Idltoluoty Qroliopjb yobjgs

L n LP n> n f() >f() ^.

Ldtb thct lbibsscrojy f(1) > 1 clgf(3) > 3.Folg n1, n3 suih thct

LP 1n1> n1 f() > 1 ^. (2.3)

LP 3n3> n3 f() > 3 ^. (2.5)

Gbffilb 1, 3 L ey

1(l) >g

1 ofl n1,

>g

3 ofl < n1.

3(l) >g

3 ofl n3,

>g

1 ofl < n3.

Whbl

OB1 >O B3.

Eut ey (2.3) clg (2.5)

f(1) > f(1) > 1 clg f(3) >f(3) > 3.

Idltrcgoitodl. Yd thbrb os ld suih fulitodl.

Gbffilotodl 2.2. Jbt L clg ` L. Gbffilb;

(l) >g

(1) of l1, . . . , l`3: l o, k : ` P o>k (lo)>(lk) ^,

>g

(l) dthbrwosb.

Fdr bcih L, wb gbffilb O B` >g

OB .

Whb sbt{OB` | L } os thb idjjbitodl df cjj olhceotbgjy blunbrcejb suesbts dfthb lcturcj lunebrs woth ct ndst ` bjbnblts.

Ldtb thct wb iclldt bvbl gbffilb c ihdoib fulitodl dl thb idjjbitodl df olhceotbgjyblunbrcejb suesbts dfL woth ct ndst twd bjbnblts. ]b icl kust idpy thb prddfdf Whbdrbn 2.7 rbpjciolm OB eyO B

5.

78


49/94

2.5 Ihdoib fulitodls dl idjjbitodls df suesbts df I

]b ndvb frdn fcnojobs df suesbts dfL td fcnojobs df suesbts dfI. Wd eb cejb tdgbffilb c ihdoib fulitodl dl c idjjbitodl Fdf ldl-bnpty suesbts dfIwb wclt cwcy td pcrcnbtrozb F.

Gbffilotodl 2.6. JbtXeb c sbt clg Fc idjjbitodl df suesbts dfX.C I-pcrcnbtrozctodl dfF os c surkbitovb ncp V ; I F, thct os, c ncp Vsuih thctF>{V() | I}.Whb ldtodl df c L-pcrcnbtrozctodl os gbffilbg ol c sonojcr wcy.

Ldtb thct ol thb prbvodus sbitodl wb hcvb usbg L-pcrcnbtrozctodls df idjjbi-todls df suesbts dfL. Ol thos sbitodl wb ncoljy idlsogbr I-pcrcnbtrozctodls.Whbrbfdrb, wb wojj fdius dl thbsb ldw clg jbcvb dut thb prbffix I of thbrb os ld

idlfusodl pdssoejb.

Jbt X eb c sbt. ]hbl wb crb cejb td I-pcrcnbtrozb c idjjbitodl F df ldl-bnpty suesbts df X, wb icl gbffilb c ihdoib fulitodl dl thos idjjbitodl cs cfulitodl f ;I X.Cs wb gd ldt rbquorb thb pcrcnbtrozctodl Vtd eb olkbitovb, ot ncy hcppbl thcttwd gostolit bjbnblts clg dfIcrb ncppbg eyVtd thb scnb bjbnblt dfF(idnpcrb thos woth thb icsb df blunbrcejb suesbts dfL, Ybitodl 2.3). ]b wcltc ihdoib fulitodl dl Ftd cssoml thb scnb oncmb td clg . Whos jbcgs td thbfdjjdwolm gbffilotodl.

Gbffilotodl 2.=. JbtXeb c sbt, Fc idjjbitodl df ldl-bnpty suesbts dfX.

YuppdsbFos pcrcnbtrozbg ey thb ncp V ;I F.C fulitodlf ;I X os cihdoib fulitodl dlF idrrbspdlgolm td V ofi

(o). IP f() V() ^,

(oo). , IP V() >V() f() >f() ^.

Whos rcosbs thb qubstodl;

gdbs thb bxostblib df c ihdoib fulitodl gbpblg dl thb ihdsbl pcrcnbtrozctodl4

Jui`ojy, thb clswbr os ld. Fdr jbt Feb c idjjbitodl df ldl-bnpty suesbts df csbtX. Ldw suppdsbV,] crbI-pcrcnbtrozctodls dfF. Whct os, V clg] crb

surkbitovb fulitodls frdn I tdF suih thct

{V() | I} > F > {]() | I} (2.9)

Yuppdsb thbrb bxosts c ihdoib fulitodl f dl F idrrbspdlgolm td V. ]b wojjprdvb thct ol thos icsb thbrb os cjsd c ihdoib fulitodl dl F idrrbspdlgolm td].Ot fdjjdws frdn (2.9) thct

I IP ]() >V() ^.

Ciidrgolm tdMCI3,3, thbrb bxosts c fulitodl ; I I suih thct

IP ]() >V(|) ^.

7?


50/94

Gbffilb c fulitodl m ; I Iey, fdr cjj I,

m() >g

f(|).

]b ijcon thct m os c ihdoib fulitodl dl Fidrrbspdlgolm td ]. ]b shdw thctm sctosffibs thb twd rbquorbnblts df Gbffilotodl 2.=.

(o). Jbt I.m() >f(|) V(|) >]().

(oo). Jbt, Iclg suppdsb ]() >]().Whbl V(|) >]() >]() >V(|) clg thbrbfdrb, f(|) >f(|).Hblibm() >m().

Yd whbthbr dr ldt thbrb bxosts c ihdoib fulitodl dl Fgdbs ldt gbpblg dl thbihdsbl I-pcrcnbtrozctodl. Ldtb thct thb scnb hdjgs fdrL-pcrcnbtrozctodls.Frdn ldw dl wb wojj usucjjy jbcvb dut thb phrcsb idrrbspdlgolm td V whbltcj`olm cedut ihdoib fulitodls.

Ol thb fdjjdwolm sbitodls wb stugy idjjbitodls F df ldl-bnpty suesbts df Iclg cs` dursbjvbs;

Icl wb pcrcnbtrozb F4

Clg fdr bcih idjjbitodl thb ncol qubstodl wojj eb;

Icl wb gbffilb c ihdoib fulitodl dl F4

]b wojj cjsd nc`b sdnb idnnblts cedut c lunebr df idjjbitodls df suesbts dfLcs sdnb df thb prddfs movbl ol thb lbxt sbitodls cjsd hdjg of dlb rbpjcibs IeyL.

2.5.3 Folotbjy blunbrcejb suesbts df I

Ijbcrjy thbrb bxosts c ihdoib fulitodl dl thb idjjbitodl df cjj suesbts dfI wothbxcitjy dlb bjbnblt. Oltuotovbjy wb idujg scy; kust sblg bcih sbt td ots uloqubnbnebr. Wd nc`b thos prbiosb, wb gbffilb c pcrcnbtrozctodl V ; {}. Whblthb ogbltoty ncp dl Ios c ihdoib fulitodl dl thos idjjbitodl.

Gbffilotodl 5.? movbs us c wcy td pcrcnbtrozb, fdr bcih ` L, thb idjjbitodldf cjj suesbts dfIwoth ct ndst ` bjbnblts. Fdr wb icl gbffilb;

Vf b` ; {1, . . . , `3}.

Rlfdrtulctbjy, bvbl fdr suesbts dfIwoth ct ndst twd bjbnblts wb iclldt gbffilbc ihdoib fulitodl. Whb prddf df thos fcit rbsbnejbs thb prddf df thb ldl-bxostblibdf c ihdoib fulitodl dl thb idjjbitodl df olhceotbgjy blunbrcejb suesbts df thblcturcj lunebrs (Whbdrbn 2.7).

21


51/94

Whbdrbn 2.8. Whbrb os ld ihdoib fulitodlfdl thb idjjbitodl df suesbts dfIwoth ct ndst5 bjbnblts.

Qrddf. Yuppdsb suih c fulitodl f bxosts. Whbl f ;I Isctosffibs

(o). IP f() >1 f() >3 ^,

(oo). , IP {1, 3}> {1, 3} f() >f() ^.

Whbl

I o {1, 3} P f() >o ^.

Cppjyolm thb Idltoluoty Qroliopjb yobjgs

I l, o IPl> l f() >o

^.

Ol pcrtoiujcr wb icl ffilg l, o L suih thct

IP 1l> l f() >o ^. (2.7)

Jbt us cssunb o > 1 (thb icsb o > 3 os prdvbl ol c sonojcr wcy).Gbtbrnolb` L suih thct

1, `< l clg 3, `< l.

Gbffilb , Iey

(p, q) >g

3 ofp, q> 1, `,

>g

1 dthbrwosb.

(p, q) >g

3 ofp, q> 3, `,

>g

1 dthbrwosb.

Whbl{1, 3}> {1, 3}. Yd ey prdpbrty (oo) dff;

f() >f().

Dl thb dthbr hclg, l> l> 1l. Hblib, usolm (2.7),

f() >1 clg f() >1.

Eut1 > 1>1. Idltrcgoitodl!

23


52/94

2.5.5 Ybqubltocjjy ijdsbg suesbts df I

Whb prbvodus sbitodl sbbns td gbstrdy cjj hdpbs df ffilgolm ihdoib fulitodls fdr

ibrtcol sueidjjbitodls dfI; bvbl fdr thb idjjbitodl df cjj suesbts dfIwoth ct ndsttwd bjbnblts wb crb ulcejb td idlstruit c ihdoib fulitodl. Fdrtulctbjy thbrb crbsdnb furthbr rbsujts. Ol Ihcptbr 7 wb hcvb nbt thb sbqubltocjjy ijdsbg suesbtsdfL. ]b icl pcrcnbtrozb thb idjjbitodl df cjj sbqubltocjjy ijdsbg suesbts dfImblbrctbg ey twd bjbnblts cs fdjjdws,

Vsi5 ; { 1, 3 }.

]b crb cejb td gbffilb c ihdoib fulitodl dl thos idjjbitodl.

Whbdrbn 2.?. Whbrb bxosts c ihdoib fulitodl dl thb idjjbitodl df sbqubltocjjyijdsbg suesbts dfI mblbrctbg ey twd bjbnblts.

Qrddf. Jbt I.]b gbffilbf() stbp ey stbp.

f()(l) >g

3(l) of ` l P 1(`)>3(`)^ clg 3(n): 1(n)

whbrbn > `P 1(`)>3(`) ^,

>g

1(l) dthbrwosb.

Yd f sblgs td thb bjbnblt df{1, 3} thct os jbxoidmrcffiicjjy thb ffirst.Ot os ijbcr thct fsctoffibs thb twd rbquorbnblts df Gbffilotodl 2.=.

Ldtb thct fdr thb fulitodl fgbffilbg ol thos thbdrbn, ldt fdr cjj I;

f() >1 f() >3.

Whbrbfdrb, fos ldt c ihdoib fulitodl cs rbquorbg ol Whbdrbn 2.8.

Whb prddf df thb thbdrbn cedvb wdr`s kust cs wbjj fdr Lol stbcg dfI. Furthbr-ndrb ot icl eb mblbrcjozbg td thb idjjbitodl df sbqubltocjjy ijdsbg suesbts dfLmblbrctbg ey ` bjbnblts, whbrb` os cly lcturcj lunebr. Hblib, fdr bcih ` wbicl gbffilb c ihdoib fulitodl dl thb idjjbitodl df sbqubltocjjy ijdsbg suesbts dfL mblbrctbg ey ` bjbnblts.Wc`olm thb ulodl df cjj thdsb idjjbitodls yobjgs thb idjjbitodl df ijdsurbs df

ffilotbjy blunbrcejb suesbts dfI (dr L). Whos idjjbitodl icl eb pcrcnbtrozbgey

Vsifb ; {3, . . . , (1)+3}.

Cjsd ol thos icsb, c ihdoib fulitodl ncy eb gbffilbg ey tc`olm thb bjbnblt thctos jbxoidmrcffiicjjy thb ffirst.

Whb idjjbitodl df ijdsurbs df blunbrcejb suesbts df I ncy eb pcrcnbtrozbgcs fdjjdws;

Vsi ; {1, 3, . . .}.

Whbrb os ld ihdoib fulitodl dl thos idjjbitodl.

25


53/94

Whbdrbn 2.31. Whbrb os ld ihdoib fulitodl dl thb idjjbitodl df ijdsurbs dfblunbrcejb suesbts dfI.

Qrddf. Yuppdsb f ;I I os c ihdoib fulitodl dl thb idjjbitodl df ijdsurbs dfblunbrcejb suesbts dfI.Ldtb thctVsi (1) >{1} clg V

si (3) >{3}. Hblib f(1) > 1 clg f(3) > 3.

Rsolm thb Idltoluoty Qroliopjb, ffilg `1, `3 L suih thct

IP `1> 1`1 f()(1) >f(1)(1) ^, (2.2)

clg

IP `3> 3`3 f()(1) >f(3)(1) ^. (2.6)

Gbffilb 1, 3 Isuih thct, fdr cjj ` L,`1 > 1 of `1,

`3 > 3 of ` `3,

> 3 of ` < `1. > 1 of ` < `3.

Whbl ibrtcoljy 1`1 > 1`1, sd ey (2.2),

f(1)(1) > 1.

Clg 3`3> 3`3, sd ey (2.6),

f(3)(1) > 3.

Hblibf(1)>f(3).EutVsi (1) >{1, 3}> V

si (3). Idltrcgoitodl!

Df idursb, thb prddf movbl cedvb cjsd hdjgs fdr thb idjjbitodl df ijdsurbs dfblunbrcejb suesbts dfL. Hblib thbrb os ld ihdoib fulitodl dl thos idjjbitodlbothbr.C sueidjjbitodl df thb sbqubltocjjy ijdsbg suesbts df L os thb idjjbitodl dfsprbcgs. Bvbry sprbcg os uloqubjy rbprbsbltbg ey ots sprbcg jcw. Whb idj-jbitodlY>

g

{ L | os c sprbcg jcw} os stroitjy clcjytoi, cs wb icl gbffilb c

surkbitovb fulitodl m ; L Yey, fdr cjj L, fdr cjj c L,

m()(c) >g

1 of1(c) > 1 c> ( jm(c) 3

m()(c(jm(c) 3)) > 1 c(jm(c) 3) >3(c(jm(c) 3) ),

>g

3 dthbrwosb.

Clg thbrbfdrb, thb idjjbitodl df sprbcgs icl eb pcrcnbtrozctodl ey

Vspr ; { L | os cgnottbg ey thb sprbcg jcw m()}

Whbrb bxosts c ihdoib fulitodl dl thb idjjbitodl df sprbcgs. Whb ogbc df thos fuli-todl os thct ot cssomls td bvbry sprbcg ots jbftndst pcth. C prbiosb idlstruitodl

df c ihdoib fulitodl dl thos idjjbitodl os movbl cs fdjjdws.

29


54/94

Whbdrbn 2.33. Whbrb bxosts c ihdoib fulitodl dl thb idjjbitodl df sprbcgs.

Qrddf. Jbt L.

]b gbffilbf() ey

f()(l) >g

o P m()(f()(1), . . . , f ((l 3)), o) > 1 ^ fdr cjjl L.

Cs m() os c sprbcg jcw thos fulitodl os wbjj-gbffilbg.Ldtoib thct fdr cjj L clg cjj l L,

m()(f()l) > 1.

Hblib, fdr cjj L, f() Vspr().Ld twd gofibrblt sprbcg jcws rbprbsblt thb scnb sprbcg. Yd, fdr cjj , L

wothVspr() >Vspr(),m() >m(). Ot fdjjdws onnbgoctbjy frdn thb gbffilotodldffthct ol thct icsb f() >f().Yd fos c ihdoib fulitodl dl thb idjjbitodl df sprbcgs.

Dl thb idjjbitodl df thb sbqubltocjjy ijdsbg suesbts df L wb iclldt gbffilbc fulitodls thct sblgs bvbry sbt td ots jbftndst bjbnblt, cs, ol idltrcst tdsprbcgs, thbsb sbts crb ldt lbibsscrojy jdictbg.

2.5.9 Bfibitovbjy dpbl suesbts df I

Ol Ihcptbr 7 wb hcvb oltrdguibg twd ldtodls df dpbl. Whb ldtodl df wbc`jydpbl os, cs thb lcnb summbst, quotb

Documents

An intuitionistic exploration of Zermelo’s Axiom