
genson padanukram

sett siddhant mein, ack ganitiya anushasana, genson padanukram yaa jey-padanukram, godel key rachanaatmak padanukram kaa ack sanshodhan haye, joe kuchh takniki kathinaaiyon koo darkinar kartaa haye joe rachanaatmak padanukram mein maujood hain. jey-padanukram sookshm sanrachana siddhant mein pramukhta seey dikhaai detaa haye, ack kshetra jiska netritva ronald gensen ney kiya thaa, jinake liye gensen padanukram kaa naam rakhaa gayaa haye.
paribhasha.
jaisa kii "l" kii paribhasha mein, maan leejiye kii diff ("x") koo "x" parr maapdandon key saath paribhaashit karane yogya seton kaa sangrah honah chaahieh
nirmaan yogya padanukram, sootra _ 2 koo paar-anant punaravritti dwara paribhaashit kiya jaataa haye. vishesh roop seey, uttaradhikari ordinals mein, sootra _ 3.
iss nirmaan mein kathinaaee yeh haye kii pratyek star aniyamit jode key gathan key tahat bund naheen haye; diye gaye sootra _ 4 key liye, sett sootra _ 5 sootra _ 6 kaa tatva naheen hoga, kyonki yeh sootra _ 7 kaa upasamuchchay naheen haye.
haalaanki, sootra _ 7 mein σ0 prithakkaran key tahat bund honey kaa vaanchhaniya guna haye.
gensen dwara ell padanukram key sanshodhan ney iss guna koo barkarar rakhaa haye aur thodi kamjor sthiti haye kii sootra _ 9, lekin yeh jodee key tahat bhee bund haye. pramukh takneek sootra _ 10 parr vanshaanugat roop seey paribhaashit sett koo code dwara kootbaddh karna haye; phir sootra _ 11 mein sabhi sett honge jinake code sootra _ 10 mein hain.
sootra _ 7 kii tarah, sootra _ 10 koo punaraavarti roop seey paribhaashit kiya gayaa haye. pratyek kramik sootra _ 15 key liye, humm sootra _ 16 koo sootra _ 10 key liye ack saarvbhaumik sootra _ 17 vidheya key roop mein paribhaashit karate hain. humm vanshaanugat roop seey paribhaashit sett koo sootra _ 19 key roop mein kootbaddh karate hain, sootra _ 20 key saath. phir sootra _ 21 aur anta mein, sootra _ 22 nirdhaarit karate hain.
guna.
pratyek upa-star "jey" "α", "aine" pargamansheel haye aur isamen "αω" + "aine" seey cumm yaa barabar sabhi crum hotey hain. upastaron kaa anukram sakhti seey "aine" mein badh rhaa haye, kyonki kissy bhee "aine"> "aim" key liye ack σ "aim" vidheya bhee σ "aine" haye. iss prakaar "jey" α "star sankramansheel aur sakhti seey badh rahey honge, aur yugman, sootra _ 23-samajh aur sankramansheel samaapan key tahat bhee bund hain. isake alawa, unake paas vaha sampatti haye joe
jaisa kii vaanchhit haye. (yaa thodaa adhik aam taur parr, sootra _ 25.)
star aur upa-star swayam σ1 samaan roop seey paribhaashit hain (i.ee. jey" "α", "aine" "jey" "β" mein kii paribhasha "β" parr nirbhar naheen karati haye), aur ack samaan σ1 achchi tarah seey krambaddh haye.  saath hii, gensen padanukram key star ack sanghanan lemma koo bahut hudd taka godel key mool padanukram key staron kii tarah santusht karate hain.
kissy bhee sootra _ 10 key liye, sootra _ 10 parr kissy bhee sootra _ 17 sambandh koo dekhate hue, usa sambandh key liye ack scholem function haye joe swayam ack sootra _ 17 sootra dwara paribhaashit haye.
praathamik kaary.
ack praathamik falan ack vn → v falan (i.ee. ack sookshm kaary joe sett koo tark key roop mein sweekaar kartaa haye) jise nimnalikhit sanchalanon seey praapt kiya jaa sakata haih
kissy bhee sett "m" key liye rudd ("m") koo sabse chhota sett honey dein jisamen "m" ≤ {"m"} shamil haye joe praathamik karyon key tahat bund haye. taba genson padanukram "jey" α + 1 = rudd ("jey" α) koo santusht kartaa haye.
pariyojana.
genson sootra _ 30 koo paribhaashit kartaa haye, sootra _ 15 kaa sootra _ 17 prakshepak, sabse badaa sootra _ 33 key roop mein, jaise kii sootra _ 34 sabhi sootra _ 35 key liye upayukt haye, aur sootra _ 15 kaa sootra _ 36 prakshepak isi tarah paribhaashit kiya gayaa haye. sookshm sanrachana siddhant key mukhya parinamon mein seey ack yeh haye kii sootra _ 30 bhee sabse badaa sootra _ 39 haye jaise kii sootra _ 41 kaa pratyek sootra _ 40 subset (α-punaravritti siddhant kii shabdawali mein) sootra _ 15-anant naheen haye.
lerman sootra _ 15 key sootra _ 43 prakshep koo sabse badaa sootra _ 39 key roop mein paribhaashit kartaa haye, jaise kii sootra _ 47 kaa pratyek sootra _ 43 upasamuchchay sootra _ 15-anant naheen haye, jahaan ack sett sootra _ 43 haye yadi yeh sootra _ 51 key roop mein ack falan sootra _ 50 kii chhavi haye jahaan sootra _ 52 sootra _ 15-aavarti haye. ack genson-shally key charitra varnan mein, sootra _ 15 kaa sootra _ 54 prakshepak sabse badaa sootra _ 33 haye jaise kii sootra _ 47 seey sootra _ 15 parr ack sootra _ 54 aparupata haye. ack kramik sootra _ 15 maujood haye jiska sootra _ 61 prakshepak sootra _ 62 haye, lekin jiska sootra _ 43 prakshepak sabhi praakritik sootra _ 65 key liye sootra _ 15 haye. 

