xflk 517.11:518.5 obobluehhtje bblhhcflehhfl h aph*methka...

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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 LlU (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 TpaacijHHHTHaa HepapxHa paajiHiHhix ypOBHefi BWHHCJIHMOCXH. Ilpn SXOM coxpanaioTca ocnoBHwe xeopeMbi xeopHH peKypciiBHbix (fcyHKUnft. IlpeflnaraeMaa cxaxta HSKOXOPMM o6paaoM npHMWKaex K xoMy >Ke HanpasneEHK). flna KaKaoro ciexHoro opaanajia c MH onpe - flenflGM T - BbllHCJIHMbie (jjyHKHHH H B 3XHX XepMHHaX CXpOHM <-O -MO - nent xnaccKiecKOft apaiJiMexHKH Bxopoft cryneHH. OrjiHHHe namero noaxo - aa ox nonxona KPHHKO aaKnioHaeTca, jcopoxso roaopa, B cnenyrameM. MH o6o6maeM onpeneneaae BHIHC^IHMOCXH xonbKO ana nncnoBbix ijiyHKUHfl. BbiHHcneHHa npon3BOAflxca aa ofiwHHbix MauiHHax TbropHHra c opaxynaMH. Opaxynbi xo»ce aanaioxca HHcnoabiMH (JjyHKiiHaMH. Bee "xpaHc- (JiHHHxHbie a4x))eK,Tbi" flocxHraroTca aa CH6T nonxonamero noaSopa opaKynoB. TSKHM oSpaaoM, OCHOBHUO noaaxna 3 name ft cxaxi>e, nq-BHAHMOMy, B

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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

TpaacijHHHTHaa HepapxHa paajiHiHhix ypOBHefi BWHHCJIHMOCXH. Ilpn SXOM

coxpanaioTca ocnoBHwe xeopeMbi xeopHH peKypciiBHbix (fcyHKUnft.

IlpeflnaraeMaa cxaxta HSKOXOPMM o6paaoM npHMWKaex K xoMy >Ke

HanpasneEHK). flna KaKaoro c i e x H o r o opaanajia c MH onpe -

flenflGM T - BbllHCJIHMbie (jjyHKHHH H B 3XHX XepMHHaX CXpOHM <-O -MO -

nent xnaccKiecKOft apaiJiMexHKH Bxopoft cryneHH. OrjiHHHe namero noaxo -

aa ox nonxona KPHHKO aaKnioHaeTca, jcopoxso roaopa, B cnenyrameM. MH

o6o6maeM onpeneneaae BHIHC^IHMOCXH xonbKO ana n n c n o B b i x

ijiyHKUHfl. BbiHHcneHHa npon3BOAflxca aa ofiwHHbix MauiHHax TbropHHra c

opaxynaMH. Opaxynbi xo»ce aanaioxca HHcnoabiMH (JjyHKiiHaMH. Bee "xpaHc-

(JiHHHxHbie a4x))eK,Tbi" flocxHraroTca aa CH6T nonxonamero noaSopa opaKynoB.

TSKHM oSpaaoM, OCHOBHUO noaaxna 3 name ft cxaxi>e, nq-BHAHMOMy, B

376 H.B.BensKHH

6onfaiueH cxeneHH, HCM a H O , Bbmep>i<aHbi B oyxe xpanHUHOHHofl TeopHH

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TEOPEMA 5. M. s o r b CL) - M o a e ji B n. n a c x e -

M b l a K C H O M (111).

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;. (3)

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. (6)

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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.