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n a n H e K o i o p o a <x - c y < 3 H o p j i a n B H o a ( H O p -M a n B H o f t ) H - y w e p a u H H y 3 3 T o r o c e u e MG T B a , o a M a H E n a e T C H ^x. - c y d H O p w a n B -H O M ( < X . - H O p M a J I B H O f t ) .
TaK KaK^QBOaHTCH K ft , TO HyuepailHH T'<<- BiraHCJIHMafl. IlO
yCJIOBHK), CymeCTBYBT $yHKIlHH Q , y , yaOBJieTBOpHKlUHe COOTHOineHH-
, (5 ) ,H O.p.$.
nony^aeu
oc - cydHOpMajiBHasTQOpeuy 4. CornacHO neuue 4, Kaasanoc-
nan HyuepaiiHH HBUHaTCHoc - raaBHoK. OOpaTHO, nycTB ̂ CCTB <x_ -nyiiepaitHH ex. - cydHOpuanBHoro ( o<- -HOpMaflBHOro)
H, cneaoBaienBHO, cymecTByei HeKoiopaa ex. -cHyMepaaan ft aioro ceueMcTsa. B cany neuuH 4, Hyue -
panHH A oc - rjiasHan H noiowy ft sKBHBaneHTHa ^ .B CMny JieiiMU 5, oTcioaa BuieKaeT, ITO ayuepartHH T- <x - cydHOpuanBHaH( <?c -
TEOPEMA 5. n y c T B po 6 i » e K T O B , H M e B m a e , c o o T B e T C T B 6 H H O ,H y u e p a u H H ^ - . y S . c c . T o r a a
a ) e c j i H ^ c y O H O p M a n B H a ^ H O p M a j i B -a a ) O T H o c H T e i B H O ot- H j3 c y f i H O p u a n B H a( H O p M a H B H a ) O T H O C H T - 6 J I B H O c C , T O / "
c y d n o p u a n B H a ( H O p a a n B H a ) O T H O c i cT e n B H o J0> ;
d ) e c n H ^ c y d H O p w a J i B H a ( H O p u a n t H a )O T H O C H T e J I B H O / 3 H / 3 C y 6 H O p M a J I B H a ( H O p -
w a n B H a ) O T H O O H i e n B H O c x i , T O ^ ~ c y C > H O p -M a n B H a ( n o p u a n B H a ) O T H O O H T e s B H O c X . .
yTBepsflemie a) sna cydHOpMaJiBHHx Hyuep^Ha.no yc -
14
, oymecTBywi o.p.$. ,L , yaoBJieTBopHBmae cooiHomeHKHM (5),(6), i.p.$. Q(m » yflosneTBOpraomafl cooTHoneHiiK) (4), H i.p.^./fc^
cooTHomeinno
(7)
CorjracHO leopeue 4, Hyiiepamifl /s «: -rnasHafl H noTOBy, cornac-HO TeopeMB I, HyiiepartHH p- jz - BiWHCJiHMa. C apyrott oiopOHH,H3(5)M (7)
ITO H TpedosanocB. OciantHHe yTBepscaeHHH nposepHioTOflHyuepaiycH IlocTa cosoKynHocTH scex p.n.u. yflosneTBOpHeT
: ona noasa OTHOCHTensHO nycToro MHOKeciBafr=». H COBO -Bcex nerycTHx p.n.M. snonae nepe^HCJiioia B szoft HyuepaqHH.
, ^10 iipn STHX ycnosaflx HOpnantHHe H cydHoptiaiiBHue ce-npyr OT_apyra TIKSH> cosepsaHHeu ocodoro
TEOPEMA 6. n y c T B o c HO B H a H H y u e p a i j H HOCc o B O K y n H o c T B 3Mt n o n H a O T H O C H i e n L H Oo C s a K i a & . T o r a a
a ) K a s c f l o e c o a e p a a m e e c r o c - c y O H O p -u a a B H o e c e w e l i c T B O o&o<— H O p u a n t H o ;
( 3 ) e C H H C O B O K y H H O C T B B C 6 X H 6 0 0 0 -
d e H H u r o f i i e K T O B H 3^72" oc- s n o j i H e n ep e i H C J i i ( u a f l , T O , a o 6 a B J i H H o <3 i> e K T ^r Kn p O H S B O H B H O U y o c - c y O H O p M a n B H O u y c e -M e a c T B y o ^ , n o n y « i H M « x . - n o p M a J i B H o e c e -M e 8 c T B o <£r, .
AoKaxeu a). Ho yenoBUD, cymecrayeT q.v»$.g(n) ,psonaH xpedOBaHioo (4). BBOAHM HOBJB Hyuepaqioo ^r ceiiettcTsanonaran
»(7 , ecnH g(/z) He onpeaeneHO .
TaK KHK ceue8cTBO oZ/" BUMMCJiiwoe, 10 ana noaxoaameit o.p.$.H,
|sI dT , e o s H / z H e onpeaeneHO
Ka noaHOTH HyuepamiH oc csesyei , ITOo.p.$. £ fa;
K xeopn BBracaBMHx cene8crB ofoeKTOB 15
^onpeseneHO
otfpasou, nan JUodHX /Z- uieeu: oiAf/7^=/Vz..no3TOHy nyo£ - BireHCHSuaH. Hs (8) nojyiaeii
1TO K
K aoKaaaTejiBCTBy yrBepsaeHUH (5), o(5o3HaiHii qepea /scex 06 - HouepOB BCOX HeocofieHHHX oO sera OB H3^7Z.- o.p.$.« ann KoiopoH _/3^ = ay^, ^^ -i.p.$.,
TpedosaHHB ( 4- ). CoBOKyiraocTB /7 HBUfleicH penepeuHcnHHOft. IIosTOMy
penypcHBHafl. BBOSHM Hyuepamoo y ceweScTBa < ,nonaran
KaK H BHBIC, nerKO npOBepneu, ITO HyuepaiiHfl ^f oC-uan. C apyroK ciopOHH, as onpeaeneHHH (^^(x.) H cooTHoineHMs (4) one
TO ecii
-, ITO M TpeOosanoci..TEOPEMA 7. n y o T L c o B O K y n H O C T t o c S s e
T o B ^2^ H w e e T a y M e p a q u i o oC TR c o s e pS H T T 3 K O f i 0 < J s e K T <?K , 1 1 0 ' C O B O K y n
H o c T £ f f lZ—{c*} OL- s n o j i H e n - e p e ^ n o s H i i a f lE C J I H H e K o i o p o e oc - n o p a a n B H o e c e i r eC T B O <^r c o s e p K H T c>t , T O c e u e f l c iB o otf0 =*& - {< }̂ o £ - c y < 5 H O p M a j i B H o e .
0(5o3Ha^un ^epe3j6 KaKyio-HMdyflB 06 - HOpiianBHyro HytaepaaKK cZH nycts -f- , ^ - o.p.$. , jyifl Koxopux
^ - p.n.u. scex 06 -Howepos scex o(5seKTOB us ~aa HHO*ecTBO /^0 BOOX Tex iHcen n, , KOiopHe ysosneTBOpflioT COOT-HomeHioo f(n)€tf , HBBHeTOH peKypcasHO nepeiHCJuniiiu H noioiiy COB-
16
nasaei c MHOxecTsou SHaieHiifl noaxoaflmea o.p.$. < f £ . OiodpameHHeif , onpesejiHeuoe
- - BiiiBCJiBuoa HyuepamieM oeMsMcTBa =^0.riOKaKeu,^TO ay-HanneTCH oC - cydHopuanBHoa. BBOSH i.p.$.
onpeneneno B
OIUOTBII TaKoe caeacTBae Teope«u 7: ecna HOpwaflBHoep.n.ii. cosepanT HaaueirbiDee UHOXBCZBO ^ , HBJiHBmeecH peKypcBB -HHVI, io ce«eHCTBO£$!r'
§ 4. 3$$eKTHBHO
A^ ycaoBBiica HasusaTB (nocTOBCKaw) HouepouoC - HyMeposaHHoa coBOKynHOCiB 7TL- ,
- odiWHaH HyuepamiH Ilocia p.n.u. 3ro ace TOCJIO /T. dyaew aa-HOMepoii (sacTBMHoM) HyiieparuiH f fseweaoTBa ^/- » onpenenae-
M0)t $op«ynofl fjc=et/( '( n^oc) , r»e K(n.,o^- $yHioiHH KJTHHH. EcnaA" (fT-fcc) onpeaenena ana scex ^C- , TO ^ HBafleiCH 06 - BHHKCHB-1108 Hyiiepamielt ceueitcTBa °Zr .
BOTHcaaiiafl HyuepaiyiH ̂ oeueMoTBa^ HasHsaeTCH cC - 3 $ -$ e K T H B H O r n a s H O f t , ecOT oymeciByei i.p.$. <^^» no -
no nouepy/z npoiOBOHtHoft cXl - Bbrancniaiofl HyuepamiB yHOMOP orf/z) o.p.$. ,
- COBO-KynnocTB Bcex SHa^eHaB $yHioiHH Kfax.) .
TEOPEMA 8 . I l O K H T H H o C - c y C 5 H O p i i a f l £ H 0 8H y i i e p a u B M B cL - 3 $ $ e K T H B H o r n a BH o f t H y u e p a u H M p a B H O C M n t H H .
nycTB ,/3 oC- 3$$eKTMBHO raaBHaH HywepaitHH ceweBcTBa OLT Bw(nJ)- cooTBeTCTBynmaH i.p.$. , nosBonniomaH no Houepy n,
Hywepaima y ceMeMcTsa & HaxoaBTi, Houep(^)$yHKiiaB,
E Teopra BUIHCUIHUUX ceueacTB ofiseKTOB 17
meft if K _/3 . nycTB o.p.$. f(zc,J ysoBneTBOpneT JTCJIOBHB .=cCf(oc). PaccuoipHM npoHSBOJitHoe IHCJIO ̂ . IIpHcoeaHHHH
K ceueftcTBy =£• , nonyiini HOBOC ceiieftciBO exf , WH KOioporo cipo-HII BHiHcmujyB Hyuepamno ^ , nonaran yx = c^ F(n^ac.) ,
F(n,'X,) OT nepeueHHHX AZ^ oOmepeKypoHBHaa H noioiiyuacna <2. HMSSM
Hycii. ctne <&- . Toraa aC=& H [<^/z-] - nouepHHyo: = oC/('^2;/z]^oeiiellcTBa oZ/'
=/3 A?^ t2? ̂ U ̂ H i c neaosaT eniHO ,
peKypcaBHyc ^ y H K U H B / t - j C y a e i i Hiiett
(9)
OCpaiHO, nyoiBa i.p.f. Q C3^) « yflOBneTBopflBmaH cooTHOineHHn (9) .
PaccnoTpmi KaKyu-HHOyai. BiWHcnHiiyio Hyiiepamho /* :<& . Tan Kan aL/C(n.fx.)c£' , TO us (9) caeayei
- noaxoaHiee c^uccHpoBaHHoe qwcno. OTcoaa BH&HO, ITO B Ka-CBOflameft lyramHii us(n,) MOSHO BSHTB lymcmuo
§ 5. CTaHaapiHue ceiiettCTBa H npeanonaneHyuepamu
£ GTaTte [2] BseaeHo cxeaynmee nomine nonHo8 Hyuepamu: ny-oC HasHBaeiCH n o n R o 8 , ecnw cymecTByei laKafl o.p.g.
, 1TO
(10)
Bcex sex rz. f SHH KOTOPHX f y H K i m f l a ^ B C D f l y onpejeneHHafl.HHa-roBOpn, HyiiepamtH oC nonHan, GCBH cymecTBye? axropHni, nosso-
no KnHHHeBCKOMy Houepy n. npoaaBonBHOlt Q.v.
peisemie ^C (aenoflBracHyjo toiicy npeoCpasosaHMH Q ) ypas-
IS A.il.Masses
H6HHH
xoro, HToOu Be cueumBara 310 noHHTue IIOJIHOTH cBHiae noHHTHew nonHOiu OTHOcmenBHO $HKCnpOBaHHoro ofoeKTa, MH 6y-aeM HasHsaTB n p e a n o n H H M H HyuepaunHMH Hywepauai, non -Hiie B CMHCJie [2 ] .
Kasaafl HyuepaoMH, nonaafl oTHocHTenBHO HeKOioporo o6i»eKTa,HB-H npeanonHOft C3 ] . B KaKHX cny^anx sepno ofipaTHoe yiBepx-
- He HBBecTHO. B [ 2] noHasaHo (o6o6meHHaH Teopeua Poasep -ca), I T O 3 K B H B a n e H T H H e n p e a n o H H H e H yU e p a U H H H S O M O p ^ H H .
TEOP3JA 9 . a y c T f c H y w e p a a n H c X , c o B O K y n -H o c T *Wt n p e a n o B H a f l . T o r a a K a m a a f l OL -u o p M a n B H a a H y w e p a u H H y 3 n p o H S B o n tH O T O oc - H o p w a n B H o r o c e w e f l C T B &£"^V?tn p e a n o n H a H K s e e o C - H o p u a n B H t i e H y -n e p a i i i i H o ^ M 3 0 M o p $ H H a p y r a p y r y .
B lacTHocTH, i c e r n a s H u e s y u e p a a K H n p o -H 3 B O & B H o r o H o p u a a B H o r o c e u e f i c x s ap . n . t i . n p e a n o J i H u e H u e a 4 y 0 0 6 0 8
; ̂ V7 - o.p. fymcmiH, yjOBneiBopfliomHa,HO, ycjiOBMB j3cc=oif(oc) B TpeCosaHHflH (9), (10), H nyciB n, -
KOTOporo $yHiamH /T^xJaoBay onpeaeneHa. HmeiiKoioporo
oiHocHTenBHO nfoc. . H a (9), (10) nonyiaeii
odpasou, g (tf(\(l,n~])) OCTB HCKOuafl HenoaBHSHafl roiKa SJM npe -oOpasosaaHH K(n.yoc) B HyiiepamiH fi u noiouy HyuepamiflyS npea-
CoraacHO teopeiie 4, see oC - HopuaJiBHHe Hyuepamjn •& oC-H noTouy apyr apyry sKBHsaneHTHHe. B cany odoOmeHHOft ieo-
Poazepca, 9KBMBaneHTHue npeanonme Hyiiepamm HSOUOP^HH B(cne-Bee cL- HOpuaaBHiie HyuepaitfiH ^ H30uop$HM.
B cooTBeTCTBHH c onpefleneHHeu A.JIazaaHa [I j, Hyuepamoo J3ceueltcTBa <£r ^?7£ ycaoBMicH HaaHBaiB c C - c i a H a a p iH o M, ecnH HyuepaqanyS od - Bmacniuiaii H
K leopan Bj-raicjiHMHX ceaeMcTB oSseKTOB. 19
, sonycKaamee oL - oTaaaapTHyB Hyaepaimio,6TCH c£- CTaaaapTHHM.
flcHO, ITO Kasaaa CTaHflapiHaa Hyuepaiyw HopiiajitHa H notouyK a x a o e ot - C T a a f l a p i a o e c e u e f t c T B O < * - -H o p u a j i t H o e . IIocneaHee yTBepzaeime sonycnaeT odpaatenae:
TEOPEMA 10. K a s a a H o L - H O p u a n B H a f l H y -M e p a u H H y 3 n p o H S B o n t H o r o c < . - H O p M a n £ -H O T O c e i i e a c T B a ° ^ 3 K B H B a n e H T H a n o s -x o 2 H « e 4 cL - c i a H f l a p T H o M H y M e p a ^ l l Hc e a e S c T B a o Z T ' , H , c n e f l O B a T e j i £ H O , n oH H T H H o 6 - H o p u a n i i H o r o H o C - c r a H f l a p T -H O P O c e u e f l c T B a p a B H o c a x B H U .
- o.p.$., TaKJie, ITO ma see*
BBOJKM Hosyn Hyuepamn) ^ oeMeftcTBa oZr , nonaraaf/z =ufi$(/T-)=d-f(g(n)) cneayei, ITO HyiiepaiiHH f oC - BOTHCHH -
Man. Kpoue TOPO,
TaKHu odpasow, Hyuepamia ^ oC - cTaaaapTHaH. CoraacHO Teopeiie *HopuajiBBHe HyuepaiyiH flif ~ BKBHsaneHTHH.
CJIESCTBHE. E c n H o c H O B H a n H y w e p a u H H ^n p e s n o n n a H , T O K a s a a n o C - H O p w a n BH B H H y y e p a u H f l y 3 K 3 0 M o p $ a a n o a x o a f l -i e 8 o C - c r a H f l a p T H o K H y u e p a u H H .
CoraacHO reopewe 10, HyuepamiHy3 aKBHsajieHTHa <^- - ciaHaapTHoM HyuepauHH ^ . CorjiacHo Teopeue 9, o6e 3TH HywepamiH asouopf-HH.
SaueTHM, qio HyiiepanMH, Hsowop^Haa oL - ciaHaapTHoJi Hywe -pam«, He odnaaHa OHTB £36 - ciaHflapTHOft, HO, KOHBUHO, dyaei c^.^-CTaHflapTHoa OTH noaxoaflieM Hyuepaiuia oC^ , H30Mop$HoK 06
TEOPEMA I I . E c j i H y S - n p e s n o j i H a H r a a s -H a H H y u e p a u H H c e i i e f t c T B a p . n . H .= / " - ^ - c y 0 H O p M a j i B H o e c e M e M c i B O a
r a e * - - a e K O T o p a n o . p . $ . , i olacTBita onyya2 aiofl leopewa, Koraa «^- coBOKynaocii. scex
P.H.M., a o^T- HeKoiopoe HopjiantHoe ceiieiicTBO, aoKaaaa A.JIaxna -
20
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JTaxnaay, Haxo-3HM ciporo nepeiHCHHiiite ceueJtcTsa KOHe^Hia uHomeciB /T, , ^
i/3- cyCHOpuanBHaH HyuepauHH <£~ i& a - n.p.f. , ann KO-raeM
(UU^T).
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FL=T^ UT^cac >O&
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10 HaMaeicH uacno p , azm Koioporo/oe/y^yO^T7 MHp€ T, B nepsoM cnyqae ann OOJIBUHX ^c HH npH naKHx </ He sunoa -
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B 0 II B H H X O d i e R T O B ^ , H M 6 K> H a H n p 8 fl -
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XHnmtOHhadeu OHaHodXaad xsoa vsjo oaioijsnec <uiHea OHwu'd esioganao ojoniitBRdoH Bd9HMdn oaoioodn sasoshBH g
•oaioamoHH saraiHSKiiEH msdaffataaBHOjfiioK 4 'n*u*d oaiogswao
HH SCHBOff SJEBK H OSlOSffiOHW SSIHIHaflHBH X89I1M OaiOJ}8n80 BOHldBffHBIO
dir aoHBOa ' OHiosaeH SH HBH HBI 'oadaiHH laBitaBioCadn' [l] HOHBIfXBff'V nHBSBHOff MCHp 8HdO$ tfOHH OH1IfOH08H S
88 H 21 BIT8d09£ 'H'Q'd 320g9n90 ptt!Bd9ll£H
K TeopnH BHIHCJIHMHX ceuefiCTB cxfoeKTOB 23
B O p . n . M . c o a e p S H T n y c T o e u n o s eC T B 0 .
ceueftcTBa <*£ ycjioBHMCfl HaausaTB <* - o n e -o a C T a n a a p T H o f l Hyuepaiyieft, ecnays <* -
H ann Bcex TZan ̂ oc/z.
3AME1AHME. E c n a o c n o B H a n H y w e p a i t H H o cn o i a a , T O K a K f l a a o c - c n e u i i a j i B H a H a y -M e p a u H H n p O H 3 B o n i > H o r o < x : - c n e u H a n B -H O T O c e M e K c T B a ^ H 3 0 M o p $ H a n o a x oS H m e S c n e u H a n B H O i ! c i a ' H a a p i H o U n y -
KaK M B § 5, BBOflHM Hyuepanioo 3^ : 7"n=/$$(n). CooTHomeHHfl (12),(13) OfipamaiOTCH.COOTBeTCTBeHHO, B yCJIOBHH
oc- cneunani>HO CTanaapTHafl HyMepaima ?TeopeMe 9, HopuanBHae nywepauHH ̂ , ?~ H30iiop$HH.
A.Jtaxnany f l j , ceueflciBO j^ p.n.ii. HasiaaeicRc n e i i H a ; i i H H M , BCJIH OHO aonycKaei cneaHantHyio c T a H -» a p T H y x> HywepauHB. npHBeaeHHoe Bbime aaueiaHMe noKasHsaeT ,UTO noHHTHH cneuHantHoro ceueMcTsa H cef-ieftciBa, oneuua«LHoro noJfaxnany, pasHocMJitHH.
Hintec^eayioniHe xeopeMH HOITM aocnoBHO noBTOpa»T cooTBeTCTByio-21M6 TeopeMH 5, 6 H 7 us npeauaymero naparpa$a .
TEOPEMA 13. II y c T B 22? ̂ /C ^ & c O B O K y n H O C -T H O l S t e K T O B , H M 6 H) 1H H 6 , C O O T B e i C T
B B H H O , H y M e p a u H H « r , 3 r 1 y 3 . T o r a aa J e c n H ^ " o c - ( c y 6 ) c n e u H a n B H a H M
ya ^ - ( - c y ( 5 ) c n e u n a n B H a H , T 0^3 <*- ( c y <5 )c u e u H a j i B H a n ;
6 ) e c n Hy3 < r - B H i n c n H M a H M ^ - ( c y Oc n e u M a n L H a H , ? " oc - B H i H c n a u a H H a a «
T O H y M e p a a a H 3 3^ - ( c y (5)
6). no ycnoBHio, cymecTByioT w.KOiophtx BunojiHeHu ycnosHH (12), (13) H
24
Kpoue Toro,
no npesnonoseHHB, cnesyei, ITO
oCpaaou, HyuepaiyiHya zr - (Cy6)TEOPEMA 14. B e a n O C H O B H H H
oc c o B O K y n H O C T H n o n H a O T H O C H i e a s -H O o f i i e K T a ^ " , T O f l o O a B J i H f l ^ T ' K n p o -• B B o a i H O i i y oc - c y C c n e u n a n B H O J i y c e -u a i t c T B y c ^ , n o n y i H M ot - c n e u H a n B H o ec a u e f t c i B O ̂ . B q a c T H o c i u , n p H y K a-a a H H O f t n n n H O i e cc , <* - c y O c n e u a a n i -a o e c e M e f l c T B O T o r a a M T o n B K o T o r -s a oc - c n e a H a z i B H O , K o r f l a OHO c o a e p-X H T (T .
IlyoTByS oc- cyOcneunaJiBHaH HyuepaiuiHc^ . Ha IIOIIHOTH °c cne-syei, UTO HyiiepanHH T ceueiicTBa^ , onpeaeneHHaH
^r , e can ^^^V He onpeaezreHOHBHHeTOH oC - BfcTCHCJIHllOH. 03 (12), (13) CJI6Sy6T, 1TO
TEOPEUA 15. I l y c T B G O B O K y n a o c T B o ( 5 i 8 K -T o B ̂ Z , H u e n ni a H H e K O i o p y a H y u e p a q H DoC , C O S e p « H T T 3 K 0 8 0 6 i 8 K T < ? K , 1 T Q
c o B O K y n n o c T B ^^— { <?tj OC-B n o n H e n e p e i a c -H H u a f l . E o n H o c - c y O c n e u H a n B H a n C O B O -K y n H o c T B<^^ ^2T c o s e p s H T ct , TO c e w e t l -o T B o&&=£s-{(y$ T a n s e c c - c y d o n e u H a B B H o e .
« : - c y ( 5 o n e i t M a J i B H a f l H y i i e p a -a H H <& H g - i.p.f. , y" - o.p.$. , ysoBneTBOpfflomae ipeCosa -HHHM (12), (13) H cooTHomeHioo jsn — <*..£ (n) . OycTB ?r - Hyuepa-UHH ceueacTBa ̂ n ¥_,& - $yinciiHH,nocTpoeHHH8 B npoueccesaieitBCTBa Teopeiiu 7.
onpeaeneHO '^v(n)»a(ft(n)))=(n)^ <* n
H, GJieaosaTejiBHO, HywepaioiH J" ^- cyCcneimanBHa.Tan KSK Hyiiepamifi flocia K coBOKynHocTJi scex p.n.u. nonaa H
ceuelCTBO scez Henycinx p.n.u. HsaneTCH Bnonne nepeqnc«miio«,TO HS
K Teoprni BHIHCJIHMUX ceueftcTB odfceKTOB 25
Teopew HenocpeacTseHHOCJIEflCTBHE. K a K a o e c n e u a a a i H o e o -e u e- ft -
C T B O p . n . M . c o a e p « H i n y c T o e MHOs e c T B O . K a x a o e H e c n e r j u a a B H o e c y 6 -c n e n n a j i B H o e c e u e f t c T B O p . n . u . n on y i a e x c H u s c o o T B e i C T B y i O H i e r o o n e -a w a n t H o r o c e u e M c T s a o i f i p a c u B a H H e un y c T o r o u a o x e c T B a .
MH XOTHM Tenep£ HS^OKHTB ocHOBHyro leopeuy A.JIaxjiaHa o cipoe-HHH cnequanBHUx KnaccoB p.n.u. H ynasaTB aeKOTOpEie ee CBHSH c pe-
B.A.ycneHCKOro.6" ceMeftcTBa ̂ KOHewHux UHOKCCTB HaausaeTCH ciporo
BOTHCJIHMOH, eOIH ^ BOTHCJHMaH H $yHKI{IIH
= UOfflHOCTB 6 (n.)
odmepeicypcHBHa. CeueSciBO KOHGIHUX uHoseciB, aonycnaiomee eiporoBOTHCJiHuyio Hyuepaiyno, HastiBaeicH ciporo nepe*iMcnnui>ai.
TEOPEMA 16. Z n R T O T O , i T o d H c e u e 8 c T-B O p . n . u . d H n o c y d c n e n , H a n B H H i i , H e -o d x o a n u o n s o c T a T o y H O , ' i T o C b i O H Oc o B n a a a n o c c o B O K y n H o o T B B s o e x p.nji.,H B f i H & B l H X C H I I p 6 ^ 6 3 I 9 1 I M I I O H O T O H H U X
x o a n i e r o C T p o r o n e p e t i H C J i n u o r o c e -U e ft C T B a K O H 6 1 H U X U H O X e C T B .
A.JIaxnaHou [l] aoKasano, ITO ceueflciBO p.n.M. Toraa H TOHBKOToraa cneimajiBHO, Koraa OHO cosnaaaei c coBOKynHOCTBB scex p.n.M.,HBnfflonHxcH npeaeJiaMH UOHOTOHHIIX nocneaoBaTenBHocTeii MHOKBCTB noa-xoanmero ciporo nepeiHcnauoro ceueficiBa KOHeiHHx MHoaecTB, coaep-sauiero nyoroe MnosecTBO. OidpacHBaH B aioft §opMynHpoBKe ycnosiie onycTow aHOxecTse, nonyMaew, BBHay Teopeu 14, 15 , leopeuy 16.
B § 7 Hau 6yaeT Hyxno HecKonBKo 6onee ofimee npeaJiomeHHetKo -Topoe HH TenepB H c$opwynHpyeu. Coraacno B.A.ycneHCKOuy [4] , ce-
p.n.M.cir aasHBaeTCH ca - nHOTHHM, ecna cosoKynHocTB scexeuy KOHG^HUX MHOSBCTB ciporo nepeiHcnaua n Kaxaas
laciB n»6oro MHoacecTsa^ H3oir coaepxHTOH B noaxoanineii KO-MHOsecTse, npHHaaneKaieu °& n coaepKanieucH B B .
TEOPEMA 17. E c u H y g - r n a B H a H H y u e p a u H Hco - n ^ O T H o r o c e u e f i o T B a p . n . u . J ^ H
n o a c e u e M c T B O x C o d n a s a e i y a - c y d c n e n , H -a n B H o S H y M s p a i ^ H e M ~s~ f T O c e u e f t c i B O
MH npHsesew noapodHoe aoKaaaienBCTso 3ToM TeopeuH, XOTH OHOn coBnaaaei HOITH nonnociBB c aoKasaienBCTBOM JlaxnaHa ynouHHyroM
26
Btirne donee lacTHoft xeopeuu.Do ycnoBMB, cynecTsyioa; o.p.$. h(n) H y.p.$. y(n) lairae, ITO
(j3h(n)j (n=o,d.,...} - c T p o r o nepeiacnauaHHOCTB B C 6 X KOH81HHX MHOS6CTB H3 <ir H
n)
cnauana, q?o coBOKynHOCTB scex KOHO^IUIX MHosecTB,npn-/T , ciporo nepe^Hcnmia. TaK KOK see KOHe^Eue
sa as xC coaepKaiCH B nocneaoBaTeniiHocTHlysA^rz;] , TO nawHHffll T6 3Hai6HHH 7Z « OTfl KOTOpHX /3 k(n) £ £ . 03 (I4-)BH-
3$$eKTHBHO,TO COBOKyHHOCTB 3Ha-
n , yflOBneTBOpHBiHX ycnoBHau: Q(h.(n)) onpeaeneHO *j3k(n) c- saseaouo peKypcHBHO nepeiHcnaMan.
OcTaeTCH noKasait, ^TO Kaxaoe KOHe^noe noauHomecTBO Z/ npoas-uHoaecTBa/56>r cosepaMTCH B noaxoanmeM KOHe^H
s e V , VeZ" f V£/9 . flonycKaeu, ITO 310 HOKHO. HycTtcnpOBaHHoe KOHe^Hoe noawHOsecTso neKOToporo B & £ , WH Koxopo-ro He cymecTByer V , yaoBJieTsopflKmero yKasaRHioiTaK KaK/SeJ^ , TO HattaeieH laKan o.p.$.
Ilycii tf ̂ - o.p.$. , coBOKynHOCTL sHaiaBHii KoiopoJl HepeKyp-CHBHafl.
< = [*("), W),...,ta)} , UTL=T,
Hueen
ofipasoii, see MHoaecTBa ceueftcTBaj HL\i o& .ECHO,HyuepanHH i—-H^ BOTHcniwaH. B caay leopenH I , sia Hyiie-
aozrsHa dHXB f> - BuiHCJiHiioa, i.e. ana nofixoAflmefi o.p. $.
Us
K Teopra ceueKcTB o<5i>eKTOB 27
4.
T m
ITOU £ r s <>;
Ho coBOKynHooTL ,peieypcHBHO nepeiHcnmia, ITO npOTHsopeiHT npeanonoxeHHOft
HoaecTsa 7" .PasyueeTCfl, leopeua 17 duna 6u npnuuu cnescTBHeu Teopeiiii 16,6u <3iuio noKasano, UTO Kasaan imBHan aywepanHH a) -nnoTHoro
oeueHcTBa 3$$8KTHBHO raaBHan. Ecna xe aro npesnonoseHHe HesepHoe,TO norao <5ti npeflCTasjiHTB HCKOTOpua HHiepec cneayiomee o6o(SmeHH8 ieo-pewu 16 :
n y c T B j3 - n p e a n o a H a a r n a s H a n n y -M e p a a H H 10 - n ^ o i H o r o c e u e i J c T B a p.n.u.^ . n o n c e « e f t c T B O j C ' 9 T o r o o e a e a o T B aT o r a a K T O H I . K O s o r a a ^ s - c y O c n e u M -a Ji t H o , K o r a a OHO C O C T O H T H 3 M H O -X 6 C T B H S K O T O p O r O C T p 0 r 0 B H 4 H C X H
M o r o c e u e a c T B a ^ K O H e q H U x U H o x e c T BH s c e x u H o t e c T B c e u e a c T B a J ^ - . H B n H -K) m M x c H n p e f l e j i a u a B O s p a c T a i o n H x n o -c n e a o B a i e n L H O c T e K K O H e i H u x M H O
3THX ycJiosHft HenocpeflCTseHHO cjieayet HS TeopeuII n 17, a flocTaTo^HOCTi, AOKasuBaeTCfl Tan ve, KaK H B COOTBCTCT-syiomeS Teopeme Jlaxnana [ I ] .
B uHTHposaHHoK CTaTto B.A.ycneHCKOro [4H Hapsay c<y - nnoTHOGTH paocuaTpHBaeTca eie ycHOBMe t
Jfenco BHSBTB, UTo AXH &> -QJIOTEHX ceueMciB ^aauKHyrocTH oiHocMTeHBHO BSHTHH npeaena BtwHcnHMofl no
sospacTaciimx p.n.ii. n noTouy Teopeua 16 aonycKaei cjieflyKiayio $opuynnpOBKy: ceuettoTBO p.n.u. Tor^a H TOJIBKO
Koraa OHO co - nnomo it T - sawcHyio.
§ 7. Bnonae nepeqacJiHUHe
coBOKynHOCTB % mieei Hyuepamos oc H & - ceMeftciBOoC5ieKTOB HS Z2t , mtecqee <x - BOTHcnHuyio nyuepamooys .Cor -nacHO BaueiaHino HS § 2, Bee noaceiieflcTBa BHjia^ n TT , rje 7" -npo-
28
H3B07IBH06 cC - B.n. CeueftCTBO, HBJIHBTCH j3 - B.n.
saua B <& . Hx HaabiBaioT c?c - npofloraaeMHua y3 - B.n.HywepauHM ft , y Koiopux see y3 - B.n. cetieftcTBa oc-npo-
, npeaciaBJiflioT ocoCuit HHiepec. HanpitMep, ann Tamix Hy -J3 - TononomH oeMe^CTBa^ cosnaaaeT c Tononoraelt, HH -
Ha oZ/ OCHOBHOM di - TononorMeM. OSHH THH aiicc Hyuepa -onHcaH B.A.ycnencKHu ( [5] , Teopeua 5 ). MH XOTHU noicasaTt,
ITO aua Teopewa HenocpeacTBeHHO cneayei us pesyjiLTaios, coaepsa -fflHXCH B § 5 H § 6.
CornacHO ycneHCKOMy, BBraHcnHMaH HywepauHHy? ceueflcTBa °Zrp.n.u. HaanBaeTCH 3$$eKTHBHO OTKpttToK, eoia Kasaoe ^3 -B.n.
c/~ COCTOHT MS BceBOswosHHx npuHaajiesamaxCTporo nepeiHcnmiofl cHCTeuu KOHe^HUx iJHOsecTB,
HCHO, MTO Kasaoe noflceMeKciBO ̂ aiorofloj!saennu. TaKMw odpasou, 3$cbeKTHBHO oTKpHTue HyitepaumixaT K ynoMHKyrouy Knaccy Hyuepamiit c npoflonKaeuuuH snojmeJIHMUMH CeMefiCTBaUH, HO, KOH6HHO, H6 UC^epnUBaiOT 3TOT KJiaCC .
TEOPEMA 18. I l y c T B H y w e p a i i J i H o C O C H O B -H O B O O B O K y n H O C T H 771- B e p X H H f l H I I O J I
H a s , H y w e p a i t H H y 3 c e M e M c i B ar n a s H a H H < X ,/- O C I B K T H H Ss a H H H e c o o T H o m e H M e u ot-^-f . T o r j aj o e q a c n o B o e M H O s e c T B O / ^ 7 .ys
(15)
KaKoe-HM6yflB KpeaiHBHoe UHOieoiBO T . CoraacHOseiuie 3 as §2, coBOKynHOCTS Ol scex oL -HaaofiKKTOB oCieKra -/
oc- raasHoM Hyuepaiiaeft -f , nonnoM OTHOcmeniHO ex. .- OflHH H3 /* - HOMepOB OCiCKTa -/ .OCOSHa^MM Mep63
o, paBHyio B Ha 7" K HeonpeseaeHHyn BHB T7 .JIOJIHOTH HyMepariHM y , Haa^eTOH o.p.$. (X-) , ann Koiopofi
oioOpaseHHe T) •• \t HyuepaaaeK ceaeiicTBa oT , cocTonmero JIHIDI, HB odieKios
CeiieMcTBO oT coaepKHTOH B ceiieftcTBe ^ , Huenmeii rnasHyjo Hywepamno j3 . KoaToay, cornacHO Teopewe -I, HyuepamiH \>ua, i.e. sun noaxoaflieft o.p.$. f(n~) wieeu
K Teopira BOTHanHMHX ceueacTB odteKTOB 29
(15) seneps
xe
HJIH 3^e Tf<=^>-f(oc)Gfl . TaKHM odpasoM. nposyKTHBHoeATT, - CBOflHTCH K fl H IIOTOMy /*7 HpOayKTMBHO.
CJIEflCTBHE. n y c t B H y w e p a u H H oC c o s o K y n -H o c T H Wt, s e p x H H H H n o n H a H , H y M e p aU H H y 3 c e w e K c T Ba^eW^ oC- r n a B H a a . T o r -a a , e c H H K a K o e - H H d y a B y 3 - B . n . c eM e 3 C T B 0 aC^^r-Q O j e p S H T H e K O T O p H H 06t-
6 K T C7t ,T
, B npoTHBHOH cflyqae cymecTBOBan dHITO Q,̂ iV-^oC, Torna MHOKeciBO ^^^
du ycaoBHHti (17) H noTouy duno CH npoayKTHBHUM, ^TO npOTHsopeiHTnpeanoJiosceHHoB peKypcHBHOft nepe^HcnmiocTH fl .
TEOPEMA 19. K a K O s a d a H H d a n a n y u e -p a u H H ^3 c o B O K y n H O C T K <& , K a m s- o e ^fl -BJI.c e w e f l c T B O oC^£- H B H H e i c H ^ 3 - c y d o n eU n a n B H U u .
IIo ycnoBHio, MHOsecTBOj3c/peKypcHBHO nepeiHcnHMO H noTouy HB-coBOKynHooTBK) soex aHaieHHll HeKoiopoft o.p.$. f(oc.) .OioOpa-p : fn. =ftj (n) OCTB BHttucaHMaa HyuepauHH cC '. Hoica -
sew, ITO OHa cydcneiwazr&Ha OTHOCHTentHO ^.p.f.^?^), onpeaeneHHOfl$opMynoft
fy (n,) onpeaeneHO =*fin=j3f(g(n>y)
H TpedosanocB.TEOPSIA 2 0 . ( B . A . y o n e H O K n f l [ 5 ] ) . r n a B H a f l
H y M e p a i i H H j S n p o n s B O H B H o r o < ^ - n n o T -H O T O c e i i e t t o T B a p . n . M . ,& H B n f l O T C H3 $ $ 6 K T H B H O O T K p U T O f t .
IlycTB oC - KaKoe-HHdo J3> -B.n. ceueiicTBO p.n.ii. CoraaoHOxeopeue 19, ceueftcTBO &C j3 - cydcneuaanBHO. OpHMeHHH Teopeiiy
17, BH3HM, wo ceuettcTBO oC aj _ naoiHoe.
30
, coBOKyiraocTBo^viBcex KOHeqHHX MHomecTB,f cTporo nepeuHcnHMa H Kaaaoe MHOsecTBO
XOTH OH OSHO noaMHOsecTBO, npraaanesamee o£^>z..Ho no ieo-peMe 18, Kasaoe MHOsecTBO, axoanmee B <& H coaepsamee XOTH 6u oa-HO MHOE6CTBO E3 oC , npHHaaaeSHT o^ . CJieaOBaT6JIBHO, oT'eCTB COBO-
scex iex HasMHOseciB uHosecis MSc/^, KOTOPHS BXOSHT B, «TO H ipetSoBanocB.B aaKHB^eHHC yKaxeu OAHH Knacc HOpnanBHux ceuetcTB p.n.u.,
c jb -BnoaHe nepe^Hc nHMHMH ceueScTsauia.fEOPEMA 21. I l y c T B c e u e f i c i B O p . n . u .<&
o . p . $ . H rL =f>ri.) (n=Ql...) - c i p o r o B U ^ H C -n H u a H n o c J i e f l O B a T e n i H O C T i . K o n e ^ H U xM H o a e c T B « 3 o Z 5 " . O f i o s H a M H M q e p e s ̂c e u e B c T B O B c e x p . n . M . , H u e t o m n x B H a73 ~* oT' , r a e a a C o oT n y c s o , j i M d o c y m e -C T B y e i i B K a a o . p . $ y H K U K H /ofit^ , i T o,/"C O C T O H T H 3 B C 6 X T 6 X M H O B 6 C T B 0 8
H e f t o T B a ° 6 ^ , K o i o p u e c o a e p a a i X O T H6 H O f l H O H 3 M H O S 6 C T B C 6 p H H 8 f>(fl) . T 0 T -
S a c e M e f t c T B O /^ n o p M a J i B H o .B cauoii sens, odosHaiHM lepes ^^ coBOKynHOcis y3 -Houepos
seex sex UHOKecTB ceiieaoTBa & , KOiopue coaepmai XOTH 6u O«HOMHOsecTBO BHaa Ooc> oceTiri- . ECHH fist- S , TO nonaraeii CS'AZ.^S? .flCHO, MTO & HBHH6TCH BBWHcnHMoa Hyuepauaeft ceueftcTBa ̂ M
(16)
flanee, Hueeu ^Tri=j^~i0C' , rae ^T - ceMeficTBO scex 3s -MHOSeCTB MHOmeCTB tiffin.) . B
(17)
Bcex ct , ann KOTOPUXHa (17) BHSHM, mo JLC^ fin.. noaTOiiy, B oray (I6),^C/^.C apyroa CTOPOHH, Kaaaoe &c(^eJfC^) , coaepMHT neKoiopoe
H noioiiy ^c^^^n- • TaKHM odpasoM,
= <^ . (18)
TaK K3K
(19)
K TeOpHH BH1HCJIHMHX CBMeMCTB CXfeeKTOB 31
lepes g(m) o.p. $yHKimio, JPH KOTOpoii AHS (19) noJiyiHM ftc^=7lQ(m.) . nosTOuy B $opuyne (18) B Kaie-CTB8 Cn. MOKHO B3HTB C(rri) H, CJteflOBaTeHBHO, OTH JIK)6orO
i.e. HyiiepauHH ^Hs Teopeu 20, 21 HenocpeacTBeHHO BmeKaeT ynouHHasmaHCH B KOH
ue § 5 HOpuanBHOCTB ceuettexBa iiHOKeciB BHaa./£nycioe, JIHOO s.n. ceiieiiCTBO p.n.M.
HocTyniuia B25.yffl.I964 p.
J l H i e p a T y p a
1. A.H.Lachlan Standard classes of recursively enumerable Sets,Zeitschr. f . Math. Logik und Grundlagenf . ,10, No 1 (1964), 23-42.
2. A.H.MajitiieB KoHCipyKTHBHue aareOpn, I, ycnexa M.H. 16, Jfe3(1961), 3-60.
3. A.H.MaJtBiieB HOHHO HyMeposaHHHe iiHOKecTsa. Anrefipa H norHKa.CeMHHap, 2 , S°2 (1963), 4-30.
4. B.A.ycneHCKHfl 0 BiraHcnHMHX onepaiyua, flAH CCCP 103, te5(I955),773-776.
5. B.A.ycneKCKHH CucTeMU nepe^HcnHMHX MHOxecis H nx HyuepaiiHK ,HAH CCCP 105, K 6 (1955), II55-II58.
