diagonal lemma

ganithashaasthra yukthiyil, diagonal lemma (diagonalisation lemma, selph-reference lemma allengil fixed pooyint sidhaantham ennum ariyappedunnu) swaabhaavika samkhyakalude chila aupachaarika sidhaanthangalil svayam-referential vaakyangalude nilanilpp sthaapikkunnu-prathyekichu ellah kanakkukoottaavunna pravarthanangaleyum prathinideekarikkaan paryapthamaaya sidhaanthangal. diagonal lemma surakshithamaaya vaakyangal godelinte apoornnatha sidhaanthangal, tarskyude nirvachikkaanaavaattha sidhaantham thudangiya adisthaana parimithamaaya phalangal theliyikkaan upayogikkaam.
pashchaathalam.
soothravaakyam _ 1 swaabhaavika samkhyakalude koottamaayirikkatte. oro soothravaakyathinum 6 enna reethiyil soothravaakyam 2-nte bhashayil oru "graph" soothravaakyam formula 4 undengil, ganithashaasthra bhaashayile oru phastt-oordar sidhaantha soothravaakyam _ 2 "prathinidheekarikkunnu".
ivide formula _ 8 ennathu swaabhaavika samkhya formula _ 9-maayi bandhapetta samkhyayaanu, ithu formula _ 2-ll anumaanikkappedunna aadya samkhyaa formula _ 11-nte formula _ 9-aamm pingamiyaayi nirvachikkappedunnu.
diagonal lemmaic oro formula formulakkum oru chittayaya maarggam aavasyamaanu _ 13 athinte godel nambar ennu vilikkunna oru swaabhaavika samkhya formula _ 14 (formula _ 15 ennum ezhuthappedunnu). soothravaakyangalkku pinneedu formula 2-ll avayude godel nambarukalumaayi bandhapetta akkangal upayogichu prathinideekarikkaan kazhiyum. udaaharanathinu, formula _ 13-neey formula _ 18 prathinidheekarikkunnu.
ellah aadima aavarthana pravarthanangaleyum prathinideekarikkaan kazhivulla sidhaanthangalkku diagonal lemma baadhakamaan. atharam sidhaanthangalil phastt-oordar peeno ganithavum durbalamaaya robinson ganithavum, r ennariyappedunna valare durbalamaaya sidhaanthavum ulppedunnu. lemmayude oru pothu prasthavana (chuvade nalkiyirikkunnathupole) sidhaanthathinu ellah kanakkukoottaavunna pravarthanangaleyum prathinideekarikkaan kazhiyumenna sakthamaaya anumaanam undaakkunnu, ennaal paraamarshichirikkunna ellah sidhaanthangalkkum aaa sheshiyund.
lemmayude prasthavana.
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avabodhajanyamaayi, formula _ 19 oru svayam-referential vaakyamaan formula _ 19 parayunnu formula _ 19 nuu property formula _ 22 undennu. vaakya formula _ 19 neey oro formula formulakkum _ 13 vaakya formula _ 25 nalkunna pravartthanatthinte oru nischitha pooyintaayi kaananum kazhiyum. thelivil nirmmicha vaakya formula _ 19 aksharaarthatthil formula _ 27 nuu thulyamalla, marichu sidhaantha formula _ 2 ll athinu thulyamaanu.
thelivu.
formula _ 29 nirvachichirikkunna function aayirikkatte
oro formula formulakkum 31, sidhaantha formula oru swatantra variable formula maathram, formula 34 allengil. ivide formula _ 35 formula _ 31-nte godel nambarine soochippikkunnu. function formula _ 37 kanakkukoottavunnathaanu (ithu aathyanthikamayi godel numpering skeeminekkurichulla oru anumaanamaan), athinaal formula _ 2-ll formula _ 37-neey prathinidheekarikkunna oru formula formula _ 38 undu.
enthaanu parayaanullath
ippol, oru swatantra variable formula _ 44 ulla oru ekapaksheeya formula formula _ 43 nalkiyaal, formula formula _ 45 ingane nirvachikkuka
appol, oru swatantra variabilulla ellah soothravaakyangalkkum _ 31: 
enthaanu parayaanullath
ippol formula _ 13 nuu pakaram formula _ 51 upayogikkuka, koodaathe vaakya formula _ 19 nirvachikkuka
appol munbathe varii ingane mattiyezhuthaam 
athu aagrahicha phalam.
charithram.
kaantorinte diagonal aargumentumaayi chila saamyamullathinaal lemma "diagonal" ennu vilikkunnu. diagonal lemma" allengil "fixed pooyint" ennee padangal kurt godelinte 1931 leeii lekhanathilo alfred tarskyude 1936 leeii lekhanathilo kaanapedunnilla.
rudolph kaarnapp (1934) aanu pothuvaaya svayam-referential lemma aadyamaayi theliyikkunnathu, athu "tii" sidhaanthathile eathenkilum soothravaakyathinu "eph" chila vyavasthakal thripthippeduthunnuvel, "tii" yill σ "eph" (° #("si")) theliyikkaavunna oru soothravaakyam "σ" nilavilundennu parayunnu. 1934ll kanakkukoottaavunna pravartthanangal enna ashayam ithuvare vikasippikkappedaathil kaarnappinte rachanakal ithara bhashayil padavinyaasam cheyyappettu. mendelson (1997, peju. 204) diagonal lemma polulla onnu godelinte nyaayavaadathil antharleenamaanennu aadyamaayi prasthaavichathu kaarnappaanu ennu vishwasikkunnu. 1937 aayappozhekkum kaarnappinte pravarthanathekkurichu godelinu ariyamaayirunnu.
diagonal lemma computability sidhaanthathile kleeninte recursion sidhaanthavumaayi adutha bandhamullathaan, avayude thelivukalum samaanamaan.

