
goodstein sidhaantham

ganithashaasthra yukthiyil, 1944 ll ruban goodstein theliyicha swaabhaavika samkhyakalekkurichulla oru prasthaavanayaan goodsteente sidhaantham, athu oro "goodstein shreniyum" oduvil 0 ll avasaanikkunnuvenn prasthaavikkunnu. laurence kirby jeff parees ithu peeno ganithathil theliyikkaanaavillenn kaanichu (ennaal randaam kramatthile ganithashaasthram polulla sakthamaaya samvidaanangalil ithu theliyikkaan kazhiyum). godelinte apoornnatha sidhaanthavum jerhard jentsante 1943 leeii nerittulla thelivum nalkia udaaharanangalkk sesham peeno ganithathil theliyikkaanaavaatha oru yadhaartha prasthaavanayude moonnaamathe udaaharanamaayirunnu ithu. paarees-haaringtan sidhaantham mattoru udaaharanam nalki.
kirby parees goodstein sequenceukalkku samanamaaya oru graph-theoritic hydra geyim avatharippichu "hydra" (lernayude puraana multy-headed hydrayude peril) veroonniya oru maramaan, oru neekkathil athinte "hedukalilonnu" (maratthinte oru shaakha) murikkunnu, athinodu chila niyamangalkkanusarichu parimithamaaya ennam puthiya headukal valarthikkond hydra prathikarikkunnu. herculese athinte thala murikkan upayogikkunna thanthram pariganikkathe, hydra oduvil kollappedumennu kirby parees theliyichu, ennirunnaalum ithinu valare samayameduthekkaam. goodstein seekwansukaleppole, kirby parees ithu peeno ganithathil maathram theliyikkaan kazhiyillennu kaanichu.
paarambarya adithara-"enn" notteshan.
paarambarya adisthaana-" "enn" "notteshan" enna aashayathinte adisthaanathilaanu goodstein seekwansukale nirvachichirikkunnathu ". eee notteshan saadhaarana bays-"enn" positional notteshanumaayi valare saamyamullathaanenkilum saadhaarana notteshan goodsteente sidhaanthathinte aavasyangalkku paryaapthamalla.
saadhaarana adisthaana-"n" notteshanil, "n" ennathu 1 nekkaal valia oru swaabhaavika samkhyayaayirikkumbol, oru ekapaksheeyamaaya swaabhaavika samkhya "m" ennathu "n" nte shakthikalude gunithangalude koottamaayi ezhuthappedunnu
oro gunakam "iai" thripthippeduthumbol, oppam. udaaharanathinu, bays 2 ll,
athinaal 35 nte bays-2 praathinidhyam 100011 aanu, athaayath. athupole, bays-3 ll prathinidheekarikkunna 100 ennathu 10201 aannu
eksponantukal thanne adisthaana-"enn" notteshanil ezhuthappettittillenn shradhikkuka. udaaharanathinu, mukalilulla padaprayogangalil 25 umm 34 umm 5> 2,4> 3 umm ulppedunnu.
oru adisthaana-"n" praathinidhyam paarambarya adisthaana-"n" notteshanakki parivartthanam cheiyunnathinu, aadyam adisthaana-"n" notteshanile ellah gadakangalum mattiyezhuthuka. thudarnnu exponentinullile eathenkilum eksponantukal mattiyezhuthuka, expressionil drishyamaakunna ellah samkhyakalum (baseukal ozhike) bays-"enn" notteshanakki maattunnathuvare eee reethiyil thudaruka.
udaaharanathinu, saadhaarana bays-2 notteshanil 35 aanenkil, ithu paarambarya bays-2 notteshanil ingane ezhuthunnu
athupole, paarambarya adisthaana-3 notteshanil 100 ennathu
goodstein sequenceukal.
oru samkhyayude "m" enna goodstein kramam "g" ("m") swaabhaavika samkhyakalude oru kramamaanu. g" ("m") enna shreniyile aadya ghadakam "m" thanneyaanu. randaamathethu labhikkunnathin, "g" ("m") (2), paarambarya adisthaana-2 notteshanil "m" ennu ezhuthuka, 2s-neey 3s-lake maattuka, thudarnnu phalatthil ninnu 1 kuraykkuka. pothuve, "m" enna goodstein shreniyude padam iprakaaramaanu
aadyakaala goodstein sequenceukal vegathil avasaanikkunnu. udaaharanathinu, "g" (3) aaram ghattathil avasaanikkunnu
pinneedu goodstein sequenceukal valare valia alavilulla gattangalilekku vardhikkunnu. udaaharanathinu, "g" (4) inipparayunna reethiyil aarambikkunnu
g" (4)-nte gadakangal kurachukaalathekku vardhichukondirikkunu, pakshe adisthaana soothravaakyathil _ 6,
avar paramavadhi formula 7 ll athunnu, adutha formula 6 ghattangalkkayi avide thaamasikkukayum thudarnnu avarude irakkam aarambikkukayum cheiyunnu.
ennirunnaalum, oru goodstein shreniyude gadakangal ethra vegathil vardhikkumennu "jii" (4) polum oru nalla ashayam nalkunnilla.
"jii" (19) valare vegathil vardhikkukayum inipparayunna reethiyil aarambikkukayum cheiyunnu
eee druthagathiyilulla valarcha undaayirunnittum, arambha moolyam enthuthanneyaayaalum oro goodstein shreniyum oduvil 0 ll avasaanikkunnuvenn goodstainte sidhaantham prasthaavikkunnu.
goodstein sidhaanthathinte thelivu.
goodsteente sidhaantham theliyikkaan kazhiyum (peeno ganithathinu purathulla saankethikavidyakal upayogichu, chuvade kaanuka): oru goodstein sequence "g" ("m") nalkiyaal, kaantor normal roopathil oordinal nambarukalude oru samanthara sequence "p" ("m") naam nirmmikkunnu, athu karsanamaayi kurayukayum avasaanikkukayum cheiyunnu. eee thelivinekkurichulla oru saadhaarana thettidhaarana "g" ("m") 0 lake pookunnu ennu vishwasikkuka ennathaanu, kaaranam "athil" p "(" m ") aadhipathyam pulartthunnu. yadhaarthathil, "pii" ("emm") "jii" ("emm") aadhipathyam pulartthunnu enna vasthuthaykk oru pankumilla. pradhana kaaryam ithaanu "p" ("m") ("k") nilavilundenkil maathram "g" ("m") ("k") nilanilkkunnu, koodaathe "p" ("m")-nte anubandha entrikal thaarathamyam cheyyumbol "g" ("m")-leeii rund amgangal thammilulla thaarathamyam samrakshikkappedunnu. appol "p" ("m") avasaanikkukayaanenkil, athupole "g" ("m") avasaanikkunnu. ananthamaaya regress vazhi, "g" ("m") 0 ll athanam, ithu avasaanippikkal urappunalkunnu.
9 enna function formula nammal nirvachikkunnu, athu paarambarya adisthaana praathinidhyam kanakkaakkukayum thudarnnu aditharayude oro sambhavatheyum aadyathe ananthamaaya oordinal nambar ω upayogichu maattisthaapikkukayum cheiyunnu. udaaharanathinu, formula _ 10.
p" ("m") enna shreniyile oro padavum "p" ("m") ("n") pinneedu "f" ("g" ("m") ("n"), "n + 1") ennu nirvachikkappedunnu. udaaharanathinu, oppam. oordinal nambarukalude kootticherkkal, gunanam, exponensiation enniva nannayi nirvachikkappettirikkunnu.
aaa soothravaakyam _ 11 ennu njangal avakaashappedunnu
aadyathethu prayogichathinushesham formula _ 12 "g" ("m") ("n") ayirikkatte, 
"goodstein shreniyude adutha ghadakam srishtikkunnathinulla "adisthaana-mattunna" pravartthanam, 
ennaal eee thalamurayile randaamathe "mainus 1" ooppareshanu munbu. 
aaa soothravaakyam _ 13 nireekshikkuka.
pinne formula _ 14. ippol nammal "mainus 1" pravartthanavum formula _ 15 umm formula _ 16 aayi prayogikkunnu.
udaaharanathinu, formula 17 umm formula 18 umm, athinaal formula 19 umm formula 20 umm, ithu karsanamaayi cheruthaanu. f (g (m) (n), n + 1" "kanakkaakkunnathin, namukk aadyam" g "(" m ") (" n ") paarambarya adisthaana notteshanil ezhuthanam, udaaharanathinu, expression formula _ 21 oru oordinal allah.
athinaal "p" ("m") enna kramam karsanamaayi kurayunnu. oordinalukalile staanderd oordar nannayi sthaapikkappetthaarikkunnathu, ananthamaaya karsanamaayi kurayunna oru kramam nilanilkkilla, allengil thulyamaayi, karsanamaayi kurayunna oordinalukalude oro kramavum avasaanikkunnu (ananthamaayirikkilla). ennaal "p" ("m") ("n") ennathu "g" ("m") ("n") ll ninnu nerittu kanakkakkunnu. athinaal "g" ("m") enna shreniyum avasaanikkanam, athaayath athu 0 ll athanam.
goodsteente sidhaanthathinte eee thelivu valare eluppamaanengilum, goodsteeninte sidhaantham peeno ganithashaasthrathinte sidhaanthamallennu kaanikkunna "kirbi-paarees sidhaantham" saankethikavum kooduthal budhimuttullathumaanu. ithu peeno ganithatthinte kanakkakkavunna nilavaaramillaatha maathrkakal upayogikkunnu.
goodsteente sidhaantham vipuleekarichu.
pakaram goodstein shreniyude nirvachanam maattiyittundennu karuthuka
adisthaana "bii" yude oro sambhavavum pakaram vaykkuka 
athu pakaram vaykkunnu. aaa kramam ippozhum avasaanikkumo?
kooduthal pothuve, "bii" 1 ", bii" 2 ", bii" 3,... poornnasamkhyakalude eathenkilum kramangal aayirikkuka.
ennittu anuvadikkuka 
m" enna vipuleekritha goodstein shreniyude padam ingane aayirikkuka
thaazhepparayunnavayaanu "bn" enna paarambarya aditharayude praathinidhyam edukkuka.
"g" "(" "m" ") (" "n" ") adisthaana" "bn" "nte oro sambhavavum maattisthaapikkuka"."
koode, pinne orennam kuraykkuka.
eee kramam ippozhum avasaanikkunnu ennathaanu avakaashavaadam.
vipuleekaricha thelivu ingane nirvachikkunnu
thaazhepparayunnavayaanu "bn" enna paarambarya aditharayude praathinidhyam edukkuka.
"g" ("m") ("n"), koodaathe aditharayude oro sambhavavum maattisthaapikkuka.
"aadyathe ananthamaaya oordinal nambaraaya ω ulla "bn".
pookumbool goodstein shreniyude "bays-chanching" pravartthanam
g" ("m") ("n") ll ninnu "f" ennathinte moolyam marunnilla.
udaaharanathinu, iff aandu iff,
appol
formula _ 22, athinaal oordinal formula _ 23 oordinal formula _ 24 nekkaal karsanamaayi valuthaanu.
arambha moolyathinte oru pravartthanamaayi sequence dairkhyam.
n" ll aarambikkunna goodstein shreniyude neelamaanu soothravaakyam _ 26 ennu nirvachikkunna vidhathilaan goodstein function, formula _ 25 nirvachichirikkunnathu. (oro goodstein sequence avasanikkunnathinaal ithu oru motham functionaanu.) formula 27-nte angeyattatthe valarchaa nirakk hardy shreniyile function formula 28, lobinteyum vainarinteyum athivegam valarunna shreniyile function formula 29 thudangiya vividha staanderd oordinal-induxed srenikalumaayi bandhappedutthi kaalibrett cheyyaan kazhiyum
chila udaaharanangal
computable functionukalilekkulla aaplikkeshan.
peeno ganithathinu aake ennu theliyikkaan kazhiyaatha oru motham kanakkukoottaavunna function nirmmikkaan goodsteente sidhaantham upayogikkaam. oru samkhyayude goodstein kramam oru turing mesheen vazhi phalapradamaayi kanakkakkaan kazhiyum; athinaal "n" enna goodstein kramam avasaanippikkunnathin aavashyamaaya ghattangalude ennathilekku "n" mapp cheiyunna function oru prathyeka turing mesheen kanakkakkunnu. eee yanthram "n" ennathinte goodstein kramam maathram kanakkaakkukayum, kramam "0" ll athumbol, kramatthinte neelam nalkukayum cheiyunnu. oro goodstein sequence oduvil avasanikkunnathinaal, eee function aakethukayaan. ennaal oro goodstein sequence avasaanikkunnuvenn peeno ganithashaasthram theliyikkaathathinaal, eee turing mesheen motham function kanakkakkunnuvennu peeno ganithashaasthram theliyikkunnilla.

