xflk 517.11:518.5 obobluehhtje bblhhcflehhfl h aph*methka...
TRANSCRIPT
Anre6pa H normta, 9, N°4 (1970), 375-405.
XflK 517.11:518.5
OBOBlUEHHtJE BblHHCflEHHfl H APH*METHKA BTOPOft CTynEHH
H.B.BEflHKHH
B 1964 rooy KpmiKe onySnHKOBan Kpaxicoe coo6meHHe Cl3 , B KOTO-
POM snepBbie 6binH BbicicaaaHbi ocuoBHbie nono»:eHHa xaK HaabisaeMOB re -
opnn MexapexypcHH. BnocnencxBHH ara xeopna nonyimia nanbHefiuiee paa-
BHTHG B paSorax paaa aBTOpoa. KpHnxe paccMarpHBaer B L l U (JiyHKiiKH, y
KOTOpbix apryMeHTH H sHaieHHfl npHHanjiexar HeKoxopoMy HananfaHOMy OTpea-
xy opflHHanbHbix MHcen. Ha Bbi6op aroro orpeaKa HaKJiaflbiBaroxca aeKOTppbie
orpaHHHGHHfl, HO ero nnHHa Mower SbiTfa CKOnfa yroAHO 6oni>iiiott (B lacxHoc-
TH, ijecH^THofl). Una xaKHx 4>yHKUnJi (fopMyjiHpyexcs o6o6meHHe sp6paH-re-
aeneBCKoro onpeaeneHna peicypcHBHocxH (CM. Q2H , cxp. 237,238). Ma-
aecTHbie npaBHna noacxaaoBKH H aaMeHbi npHMenaioTCH B C 13 K pasencT -
conepJKamHM opnHaa^bi. Hapaay c STHMH npaBHjiaMH BBOAHTCH eme
aaanor axaMii-onepaunn (CsD, cxp. 254), Koxopaa, KBK Hasecx-
HO, arpaex ocHOBononaraHDinyM pojifa B xeopnn rnnepapa(j)MexHHecKHx (})yHK-
UHft. FIpH yBejiHneHHH flnHHW ncxoziHoro opzumajibHoro orpeaKa nonyiaexca
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coxpanaioTca ocnoBHwe xeopeMbi xeopHH peKypciiBHbix (fcyHKUnft.
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aa ox nonxona KPHHKO aaKnioHaeTca, jcopoxso roaopa, B cnenyrameM. MH
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opaxynaMH. Opaxynbi xo»ce aanaioxca HHcnoabiMH (JjyHKiiHaMH. Bee "xpaHc-
(JiHHHxHbie a4x))eK,Tbi" flocxHraroTca aa CH6T nonxonamero noaSopa opaKynoB.
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376 H.B.BensKHH
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H H f l e x c o M (ana paanHHeana opaKynos). FIpH STOM B
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IIpH 3TOM ycJTOBHMca, Hxo OTBex t . He onpeaeJieH, GCJiH A/ nojiyHaex
oxKaa Ha /z- -ft sonpoc HHH ecnH A/ aanaex Menee HSM /t sonpocos.
nycxb
ecTfa nocJieflOBaxenfaHocxb oxsexoa, nonyneHHbix MaiuHHoit
O6o6meHHbie BHHHcneHHg H apaftMexHKa BTOpoit cryneHH _ 393
Zlajiee, xamaaa HS MauiHH /V H e(\f) conocxaBJiaex c aaHHbiM ap —
ryweHTOM jc HBKOTOpoe BbiHKcneHHoe eio aHaneHiie (BOSMOKHO, neonpe -
neneHHoe). Oycxfa STH 3HaneHHH 6yayT u)r K uJ , cooxBexcxBeHHo.
Ha (S), (8) nerKO cneayex, HTO ecaiH ana scex /z. tl^ ^. ^f^ , TO
t-t^1 !±L ^z • OCTan°cfa npoaepHTb, HTO HyjKHOe coBnaaenHe OTBGTOB aeftcx-
Mecro./ ^no HnnyKUHH, HTO tt — W^. nna x^-n. . AonycTHM cna-
nana, HTO A/ aanaeT He Menee /z.4- / Bonpocos, H nycTb ee /z.4- / -fl BO-
npoc paseH •£&, y> a noflaerca na Bbixoa i, . Torna, B cmiy (7) , (8), ( 9 ) ,
/T+/ -ft Bonpoc MauiHHbi e.(M) 6ya.er^e(ac),y> H OH 6yaeT nonaa na
Btixon <p(i) . Ha ( 10) cnenyeT, HTO
Morno cnynHTBCH, HTO A/ He aaaaeT nA-t -ro sonpoca. Ho xoraa, B cmiy
(5), e.(fiS} TOJKG He aanaeT /t+ / -ro Bonpoca. rioaxoMy CHOBB c / . a? w f ^ ./Vrf /ITT
TeM caMbiM (iii) aoKaasHo.
O6paTHMca, HaKOHeu, K noKaaaxenbCTBy xoro, ITO (JiyHKuaa e(fr/J yao -
naineft TeopeMe. FlycTb
cyTfa CHcxeMbi opaKynoB, nocxpoeHHwe ana HyMepaanJi \ * B cooTBeT-
CTBHH C § 3. AOKaKGM, HTO anH A •£ II V II HMeeT M6CTO:
, y>) ^ C< « <:«;, y > ). ( 1 1 )
no HHoyKUHH, HTO:
Civ) fljia Bcex opaHHanoB, MeHbiiiHx nexoTOporo /i , cooTHOineHHe
(11) BbinonneHO.
Kaxabift H3 opaxynoB ff^ H ^ CTPOHTCH nocpeacTBOM HeKOxopoS
^'^fl!^ -onepanHH (npH noaxonarnHx /? H *r ), npHMenaeMofi K cosoKyn -
HOCTH npeAiuecTByiomHx opaxynos cooTBexcTByromero ceMeftcraa. CpaBHHaaa
STB onepauHH H Hcnonbaya (.iv) , ySeanMca, HXO (11) BbinojiHsexcsi H ana
paccMaxpHBaeMoro A . HUJKB MU 6yaeM npHMeHflTb ofioaHaneHHa HS § 2,
MOflH(})HUHpOBaHHbie OHeBHflHblM o6pa3OM npHMeHHTSJIbHO K
'=.\?~ A , /7A={£, ]V,i^ A] H x.n.)
no V aoxa>KeM, HTO ana scex A/,cc;
394 H.B.BenaKHH
a, B cany (iv) H onpeneneHHa ffi# ^ -onepauHH, H 6ynex
( 1 1 ) . EflHHCTBeHHblH HSTpHBHaJIfaHblfi CnyHafi, KOTOpblft HaflO
npH aOKaaaTejifaCTBe (12), - STO nepexon OT jf K y+ / (CM. (B-) ,
H3 § 2). HxaK, npHMBM euie OflHO HHnyKTHBHoe npeflnonojKeHHe:
(v) flna HeKOTOporo •{ HMeer Mecro cooTHOineaHe (12).
B cany (4) , JXCA/^ £ /?, -*—*~^(€CM)) £Z /}. . FIosTOMy, ecnH
He ^fl, , TO &. '" fcMfZ'?) H & (^eCMjjo) oflHOBpeMBHHo He onpene-* • » . ' • • A .
nenw. PaccMorpHM Tenepb TaKyro MauiHHy M , HTO Cf(.M) C A . Ilono—
er, ITO CM H O^ yaoBnexBoparoT ycnoanio (10). Paccy*aaa npHMepHO ran
«e, Kax npH aoKaaarenbCTBe (iii) , nonynaeM
OrcKma, B cnjiy (B.) us § 2 cnenyer, HTO (12) BbinonHHercH ana
TeM caMbiM MH noKaaanH cooTHomaHHe (12), a BMBCTB c HHM H (11). Ha
(11) H (iii) cjienyeT, ITO {A/j ( f )c^c; a^ { e ( /V)} *\x) . TeopeMano-
xasaHa
§ 6. CoBMecTHaa cacTSMa
PaccMOTpHM CHCTemy HyMepauHfi { \ ) , rae f npo6eraeT see CH&T-
HHe opuHHajiw, a n v^ll HanaercH npenejifaHbiM opnHHanoM >•£" .
OFIPEflE/IEHHE 6. CncTeMa \_^v} HaawBaerca COBMBCTHOH, ecnH npn
2" ^ Z^ acaKaa (JiyHKUHa, (o6o6uieHHo) BbiiHcnHMaa OTHOCHTenfaHO HyMepaiiHH
Vj. , BblHHCJIHMa TBKiKe OTHOCHTenfaHO V*- .' ri
CyinecTBOsaHHe coBMecTHofi cucTeMbi nyMepauHfl 6yneT noKasaHO HH«e.
flna a-roro HBM npHflerca npeABapHTenfaHO cnerKa MonH$HHHpOBaTb reopewy 3.
OycTb y3 - Kaxofi-HHSyab cieTHbifl opnHHan. ByaeM roBopHTfa, HTO uyviepa-
una j t oziHOCBOAHMa K HywrepauHH v^ co C f l B H r o M / 5 , ecna
cymecTByeT oSmepeKypcHBHaa pasHosHaHHaa ({jyHKUHS <f> raKaa, ITO ana
Bcex K yS-f V,Z= ^ <P(x) . IlycTb Tenept, iJiyHKiiBa e(/^fj nocTpoeHa, HC-
xoaa K3 3Toft (JjyHKHHH 'f , TOHHo ran »e, K8K B npeflbinymeM napaTpa^e.
HEMMA 5. { M\^ (x) t* { e(M)} (x).
flOKAS ATE/1 bCTBO. flna nroSoft MauiHHbi M HMSOM (no nocrpoeHHio
(*< I V, W -
O6o6meHHfaie BWHHcneHHa H apmJjMexHKa BTOpofi CTyneHH 395
Orciofla, paccy>Kn.as TBK JKG, KBK npn flOKasarejifaCTBe reopeMw 3, nonynaeM:
Ha 3Toro cooTHouienHH (B ciuiy (iii) HS § 5) BHTeKaex HCKOMOG yrsep -
TEOPEMA 4 . C y m e c T B y e r c o B M e c T H a a c H -
C T B M B H y M e p a u H f l .
flOKASATE/IfaCTBO. Pa3o6i,eM MHOJKSCTBO HarypajibHbix HHcen ^V na
HHCJIO 6ecKOHeiHfaix HenepeceKaMfflHxca peKypcHBHtix
M = U Jf .
<}>yHKUHa, oxo6pa>Kaiomaa BaaHMHo OZIHO -
-V na JVrrl (,m= O, -f, ... ).
HcKOMyro CHcreMy HyMepauHft 6yneM crpoHTt c noMoaifaio rpaHC({)HHHT -
HOft HHAyKUHH. flonyCTHM, HTO flnfl K3>KflOrO O «= Z Mbl yMC6 HOCTpOHTIH Hy -
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flHMH K HyMepailHH \ .,. C HeKOTOpbIM CflBHrOM.L
BoabMeM renept npoHaaonbHyio nyMepauHio V^ , ana KOTOpoft ''
ecrfa npenejibHtifi opaHHan ^.T ' . BoafaMeM xaKace npoHasojifaflyio
6 O"*-) , oxo6pa>Kaioinyio -V Ha MHoacecxBO opflHHajioB : t . flna
i ll Vg II , ecnH 0 <: zr ;
*V^ , ecnH f t- .
OnpenejiHM HyMepaimro v1 . cneayiomuM oSpasoM. riycxfa ze. J^m ana KBKO
roporo m . Torna:
* -//3 + "J y/ eui r -*-/n
PaccMOTpHM nio6oe C •£. t , H nycTb rn. TaKoao, HTO ff(m)= G . Toraa,
no nocrpoeHHio Vj. , v onnocBoaHMa K v*^ co caBHroM 2< lbQ , npHies/ &(L)
CBonamefi <j)ynKUHeft asjiaeTCH ^f
B cany neMMw 5, nocrpoeHHaa raKHM o6pasoM cHcreMa HyMepaunft [v^J
coBMecTHa, H TeopeMa
396 H.B.EensKHH
H a H H H a a c a T o r o M e c x a H no K O H -
u a c x a x b H M H 6 y a e M n p e n n o n a r a x b ,
H x o 3 a ( j ) H K C H p O B a u a H e K O T O p a a c o B M e c T-
Haa c H c x e M a H y M e p a n a f t ^ r • KpoMe roro, MH
noTpe6yeM, ixoSw ana <$ < £ v? 6bwa oaaocBoaHMa (co caBHroM) K
nocpeacxBOM HeicoTopoft 4>yHKUHH < ? _ H BbinonHHnocb HepaseHCTBO:
2
, cucTGMa HyMepauuft, nocrpoeBHaa B AOKaaarenfaCTBe xeopeMW 4,
flonojiHHTent>HOMy Tpe6oBaHHK>.
OnPEflE/IEHHE 7. 4>yHKuHa HaabiBaexca f -BbiHHcnHMoft, ecnH oaa
o6o6meHHO BbiHHcnHMa OTHOCHxenbHo HywepauHii v'— . AHanornHHo onpene-t*naerca X -paapeuiHMOCTfa ana npeflHKaxoB.
MM 6yjieM ynoTpe6naTt> orpaHHHeHHbie (jiyHKUHOHa^faHwe KBaHTOpw BHna
Voc H 3 'oc , noapaayMeBaa, HTO nonKBanropHaa nepeMeHHaa irpofieraer co-"c t
BOKynHocxfa scex oaHOMecrHbix BCioziy onpeaeneHHbix T -BbiiHCJLHMbix ijiyHK-
UHft.
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u i H M w f t n p e a H K a r . T o r n a :
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flOKASATE/IbCTBO. Ha nesofl nacTH HCKOMOU sKBHBaneHTHOCTH rpn-
cneayex npasaa nacxb. floKa>KeM o6paxHoe. Ilycxfa ana HesoToporo
cnpaaennHBo:
OpeflHxax Q. "? -paapeuiHM; noaxoMy npH HBKOTOPOM A. •* 11 tf_ )l OH aana-
exca (VpiAj -paapeiiiHMbiM. CornacHO cneacTBHro 2, Ha (2) Btixeicaex, HXO
cymecxsyex £V.j.,.X-MJ-BbiHncnHMaa uent Qffi , ana Koropoft cnpaseanHBo:
Tax K3K offl 2T -BHHHcnHMa, xo ana aaHnoro OL
r
TeM caviHM cneacxBHe
C/IEflCTBHE 4 . E c n H ex ZT , T O M H O X B C X B O
O6o6meHHbie BbiHHcjieHHH H apHjiMerHKa BTOpoft cryneHH _ 387
B c K> a y o n p e a e n e n o } a B n a-
e T c a t - p a a p e u i H M M M .
flOKA3ATEflbCTBO. B cuny neMMbi 5,
( I , , (^J <z I I Z j
{ Z |
Bcrofly onpeneneno I =
scrony onpeneneHOJ , (3)
rne £03? nocrpoena, HCXOA« HS cBonnnieft (JiyHKUHH S j. . Ha (1) cnenyer,
HTO ana HexoToporo \ II \>zl\
} C-«9 Bcrofly onpefleneHoV £ ?. (4)
Ma (3), (4) c noMomfajo neMMBi 3 nonynaeM, HTO paccMaTpHBaeMoe
CTBO {\^, Aj -nepeiHcnHMO H, cnenoBaTejibHo, {^,A.+ ^} -paapeuiHMO.
CnencTBHe
§ 7. CTa6HjiH3auHa
. Pac-
3<x.0 Voc,
rue 6?^? ecTfaj'V npn /z, HBTHOM H V-^ np« /^- HenetHOM, aA.^,...,
cyTb cneTHbie opflHHanbi. MHOWBCTBO Tex a. , ana KOTOPHX (1) HCTHHHO,
nepea:
ITpeflnonowHM, HTO A,0,...,A-n onHOBpeMeHHo BoapacxaiOT no neKOTOpOMy aa-
KOHy, T.e. A-=A^Cr ; Ci.=O,...,n.) , rae napaMeTp -ZT npo6eraeT Bee
CHGTHbie opflHHanbi H see Afc- C2\ cyTt MOHOTOHHbie (JiyHKUHH. Torna nnxep
nperauHH (j>yHKUHOHanbHbix KsaHropOB B (1) 6yaex MGHHTIJCH c H3M6H6HHeM
f . CoOTBeTCTB6HHO 6yn6T MCHflTfaCH H MHOJK6CTBO (2). Mbl SaHHTepeCOBa-
Hbi B raKOM aaxone pocTa A-j , npn KOTOPOM (2) MO»eT jiHiiifa pacuiHpHTfaca ,
Korna Z" BoapacTaeT. Torna, B cnny HecnexHocTH MHoxocxsa cnexHbix op-
aHHanos, MHOMOCTBO (2) Korna-HH6yAb cxa6HnH3Hpyexca.
3xH paccMOxpeHHH noncKaawBaiox aienyromyio aanany, peuieHHio KOXO —
poft H nocBflineu nanHbifi naparpa4>. Mw XOXHM nocxpoHTb
(3)
398 H.B.EensKHH
jrna xoxopofi BbinonHeHw ycnosHH (i) , (ii), (ill) ,yica3aHHbie HHJKO B
7. CoAepJKaxenbHHfi: CMbicn SXHX ycnoBHii MOXHO npeflBapaxenbHO no-
Tax. fljia npoHasonbUbix n K <5 H npoHSBpjibHoro o -paspeuiH -
Moro n.+2 -MecxHoro npenHKaxa & oSpaayeM (JiopMyny (1). Toraa HCKO-
Mbie 4>yHKUHH A-t' (z) , oSecneHHBaroniHe MOHOXOHHOS KSMeHeuHe cooxBercxBy—
jomero MHOwecxsa (2), MO*HO nonynnxb KBK cynepnoaiiuHH (Jiymcmifi HS no-
cnenosaxenbHOcxii (3). BHA SXHX cynepnoaHuaft 6ynex aaaHcexb xonbico ox
n , a paciiinpeHHe MHOwecxaa (2) rapaHTHpyexca nna 2~>e .^Cc-^ecxb
MOM6HX CXa6njIH3aUHH 3XOPO MHO)KeCXBa.
BaeaeM Henoxopwe BcnoMoraxenbHwe o6oaHaieHHa, cBasaHHwe c no —
cnefloaaxenfaHOCTbKi (3). FlycTfa
flanee, nna/z.,* / H npOHSBOjibHoro /i-HJ -MecTHOro npeflHKara G nono-
>KHM:
/1EMMA 7 . C y m e c x s y e x n o c n e n o B a x e n b —
H O C T b O p f l H H a n b H W X ( j j y H K U H f l ^ C t ^ (TV* 4,2,...)
x a K a a , i x o : ( i ) a c e _ c x p o r o M O H O X O H -
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ro cs - p a s p e u i H M o r o n.+ 2i - M e c x H o r o n p e -
n e x a x a - C (_n,z* f) H M e e x M e c r o :
(ii) E o j i H e < r 7 < Z ^ , xo 3 %*' S S£v',
(iii) Ben. r^^fe;. , TO 'r = ' ^ .
flOKASATS/IbCTBO. FIocxpOHM cHaia.na (J)yHKHHio «^ . flycxfacs— npo -
ciexHMft opflHHan H nycxb £ (a , U,v) - Kaicoft-HiJByflb <5 -paa-
peuidMbift npenHKar. flna npoHaso/ibHoft iiicnoBOit (JiymcnHH oCCx) HOJIOJKHM:
ecjiH oc - Z -BwiHcnHMaa (jiysKUHH C^y-o) , xo -^ ecxb
_ O6o6meHHbie BbiHHcneHHa H apmfrMexHKa Bxopoft cxynenH _ 398
paspeuiHMHft npeanicax. Ha (4), (5), (6) wieayex:
,^— — «e-S£ [>, Z\ »-_3x: Vfij3jce(a<£(jc),fi(x))
flna T >T? H ana BCaKOft £" -BblHHCJIHMOft (JiyHKIIHH CC C IIOMOIUblO
cneacTBHH 3
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HaKOHeu, OIBBHAHO, (nnsi £^>r );
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FIpaByio lacTt. HwnnHKauHH (9) MO>KHO nepermcaTb B BHne ae. £:> ' *
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opnHHan Zp , nasHHaa c Koxoporo MHO«ecTBO £& CTa6HjiH3HpyexcH,r ^v ' ' ' ei'.e.T.e. ana ^ ^-^ -Sg = S . Tax KaK HMeexca Bcero nnuifa ciexHoe
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9>, (0) > v' . (11)r c
Ha (10), (11) cneayer, ixo ana /r=./ cnpaBeanHBo (iii) . ECJIH Mbi ao-
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400 H.B.BejiHKHH
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ana He6 (iii) H (i) . BoabMeM npoHSBontHbifi opaHHan <s
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MBCTO (iii) . ConocTaBJiaa STH c()aKTbi,
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J7cc*"
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a&g£'T* . OosTOMy H3 (14), (15), (16)
_ O6o6meHHbie BbinncneHHa H apHifrMeTHica BTOpoft cryneHH _ 401
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uiHMoro £ cyujecTByer CHerHbifi opaaHan •zr^ , HaiHHaa c Koxoporo
MHO«ecTBo £> g' cTa6nnn3HpyeTca. Bo-BTopbix, cyiaecTByer cnSTHbifl op-
an«an fn xaxoft, iTO'Eny- Z:^ ana nio6oro t=> -paspeuiHMoro ^
Bw6epeM <p TBK, HTo6bi ana Bcex e> 6bmo ^L(e>) >• ~C^ • Torna ana
naHHoro n. 6yneT BbinonneHO (iii). flononHHrejifaHo norpeSyeM, HTo6bi
f- Sana crporo MOHOTOHHa H MawopnpoBana see ^YHKUHH *P,..,, <f> _
Torfla nna ~n. 6yaeT BbinonHeHO (i) . TGM caMbiM neMMa noKaaana.
§ 8. Moaenfa apnifiMeTHKH Bxopoft cryneHH
PaccMOTpHM (JiopMajifaHbiH asbiK oC^ , npe ACTaBnaioiuHfl co6ofi paaHO -
apH<J>M6THHecKoro aswKa BTOpofi cTyneHH. B o£^ HMewTca nepe -
flByx THHOB: npeaMernbie -sc,^/, . , . H (JiyHKUHOHanbHue or, /g, . . .
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nepeM6HHbie npoSeraiOT MHOWGCTBO HaTypanfaHbix HHcen, anaKH
HHTepnpeTHpylOTCa B COOTB6TCTBHH C HX o6bIHHbIM nOHHMaHH6M
B apHi})MeTHKe HaTypanfaHbix HHcen, a 4)yHKUHOHanb.Hbie nepeMSHHbie npoSe -
raror MHO«ecTBO acex oanoMecTHbix Bcroay onpefleneHHbix HHCJIOBWX
KaK o6biiHO, TepMaMH Hamero asbiKa £ HBJiaioTCH: 0,1, npeflMeT-2i
Hbie nepeMGHHbie H BbipaxeHHa BHna ^,~t" ^2 , ^-,'^2 t*^ ) , rae oc -
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TapHbie (JiopMynbi cyTfa pasencTBa TepMos. OcTantHbie (jjopMynbi cTpoarca HS
aneMeHTapubix nocpencTBOM onepauHfi /IOPHKH npeflUKaroB, npnneM paapeuia-
eTca HaaeiuHBaTfa KBaHTOpw Ha nepeMGHHbie o6onx THHOB.
PaccMOTpHM Tenepb- (JwpMajifaHyio cncTeMy^-^ , HasecTHyio IIOA
HHGM ap«(j)MeTHKH BTOpofi cTyneHjj. STB CHCTeMa nonfaayexca aawKOM at. H
cocTaBJiena H3 nocTynaroa cneAyroimix Tpex BHAOB:
(1) AKCHOMW H npaanna BbiBOAa KJiaccHnecKoro Hc<jHcneHna npeAHKa -
TOB BTOpOfi CTynOHH.
(FI) AKCHOMbI apH({)MeTHKH H3Typa.TbHbIX
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nonHflerca ana BCGX n. . Teopevia
/ l a T e p a x y p a
1. S.KRIPKE, Tranefinite recursione on admissible ordinals, I,
J. Symbolic Logic,29.N3,(1964),161.2. C.K.JCflHHH, BBeneHHe B MeraMaTeMaTHKy, \\I\, 19S7.3. H.ROGERS,Jr, Theory of Recursive Functions and Effective
Computability ,Mc Graw-Hill Book Company.
6 anpena 1970 r.