
maathrukaa sidhaantham

ganithashaasthra yukthiyil, aupachaarika sidhaanthangalum (oru ganithaghadanayekkurichulla prasthavanakal prakadippikkunna aupachaarika bhaashayile vaakyangalude oru shekharam) avayude maathrkakalum (sidhaanthathinte prasthavanakal nilanilkkunna gadanakal) thammilulla bandhathekkurichulla padanamaanu maathrukaa sidhaantham. oru sidhaanthathinte maathrkakal ethra valuthaanu, avayude ennam, vyathyastha maathrkakal thammil ulla bandham, aupachaarika bhashayumaayulla avayude edapedal enniva anweshikkunna vashangalil ulppedunnu. prathyekichum, oru sidhaanthathinte maathrkayil nirvachikkaan kazhiyunna settukalum atharam nirvachikkaavunna settukalude parasparam bandhavum model sidhaanthakar anweshikkunnu.
oru prathyeka vibhaagam enna nilayil, 1954-ll prasidheekaranathil "modalukalude sidhaantham" enna padam aadyamaayi upayogicha alfred tarskyude kaalathaanu model sidhaantham.
1970kall muthal, saharon shellayude sthiratha sidhaantham eee vishayathe nirnnaayakamaayi roopappeduthiyittundu.
proof sidhaantham polulla ganithashaasthra yukthiyude mattu mekhalakalumaayi thaarathamyappeduthumbol, model sidhaantham palappozhum aupachaarikamaaya kaadinyavumaayi bandhappettathalla, koodaathe classical ganithashaasthravumaayi aathmaavil kooduthal aduthirikkunnu.
thelivu sidhaantham vishudhanekkurichaanel, maathrukaa sidhaantham ashudhiyekkurichaanu" enna abhipraayathinu ithu prerippichu.
beejaganita, diophantine jyaamithikalilekkulla maathrukaa sidhaanthathinte prayogangal classical ganithashaasthravumaayulla eee saameepyathe prathifalippikkunnu, kaaranam avayil palappozhum beejaganita, maathrukaa-saidhaanthika phalangaludeyum saankethikavidyakaludeyum samyojanam ulppedunnu. thathfalamaayi, ardhaparamaaya model sidhaanthathinu vipareethamaayi, proof sidhaantham vaakyaghadanayude swabhaavamullathaanu.
maathrukaa sidhaantha mekhalayile ettavum pramukhamaaya pandithochithamaaya samghadana pratheekathmaka yukthikkayulla samghadanayaanu.
avalokanam.
eee peju ananthamaaya gadanakalude phinittari phastt oordar model sidhaanthathil shradha kendreekarikkunnu.
vishayathinte charithratthil ettakkurachilukalla oru modalinullile nirvachikkaavunna settukalude classinu vipareethamaayi oru sidhaanthathinte modalukalude classinu nalkunna aapekshika oonnal, koodaathe rund dishakalum yathaakramam 1973,1997 muthalulla sookshumamaaya swabhaavasavisheshathakalaal samgrahikkappedunnu
saarvathrika beejaganitam ganithashaasthra gadanakaleyum yukthisahamaya sidhaanthangalude yukthiyeyum soochippikkunnu; oppam
nirvachikkaavunna logical soothravaakyangal oru feeldile enangalkku samavaakyangal enthaanennu sajjamaakkunnu.
ennirunnaalum, modalukalude klaasukaludeyum avayil nirvachikkaavunna settukaludeyum parasparabandham athinte charithrathiludaneelam model sidhaanthathinte vikasanathinu nirnaayakamaan. udaaharanathinu, oru nischitha kardinaalthayil sidhaanthangale avayude modalukalude ennam anusarichu tharamthirikkunnathinaayi sthiratha aadyam avatharippichappol, nirvachikkaavunna settukalude jyamiti manasilakkunnathil sthiratha sidhaantham nirnaayakamaanennu thelinju.
phastt oordar model sidhaanthathinte adisthaana ashayangal.
phastt oordar lojik.
booliyan connective formula _ 1 vazhiyum quantifier formula _ 2 allengil formula _ 3-nte prefixing vazhiyum "r" ("f" ("x", "y"), "z") allengil "y" = "x" + 1 polulla aattomik soothravaakyangalil ninnaan oru phastt-oordar "formula" nirmmichirikkunnathu. oru veriyabilinte oro sambhavavum anubandha quandifierinte paridiyilulla oru soothravaakyamaanu oru vaachakam. soothravaakyangalkkulla udaaharanagal φ (allengil φ (x) aanu, paramavadhi x ennathu φ-ll oru anboundu variable aanenna vastutha adayaalappeduthunnu), si inipparayunna reethiyil nirvachichirikkunnu
(samathwa chinnathinu ivide eratta ardhamundennu shradhikkuka.) atharam soothravaakyangal engane ganithashaasthra ardhathilekku vivarthanam cheyyaamennu avabodhajanyamaayi vyakthamaan. swaabhaavika samkhyakalude σsmr-ghadana soothravaakyathil, udaaharanathinu, "n" oru prime nambaraanengil maathram "n" enna moolakam φ enna soothravaakyathe thripthippeduthunnu. si enna soothravaakyam irreducibilitye samaanamaayi nirvachikkunnu. samthrupthi bandha soothravaakyam _ 6-nuu tarsky oru karsanamaaya nirvachanam nalki, chilappol "sathyathinte nirvachanam" ennu vilikkappedunnu, angane oraal eluppathil theliyikkunnu
vaakyangalude oru settu formula _ 9 neey (phastt-oordar) sidhaantham ennu vilikkunnu, athu settile vaakyangale athinte svayamsidhaanthangalaayi edukkunnu. oru sidhaanthathinu oru "model" soothravaakyam _ 10, i undengil athu "thripthikaramaanu".iii. settu formula _ 9 leeii ellah vaakyangaleyum thripthippeduthunna oru ghadana (uchithamaaya oppulla). oro vaakyavum allengil athinte nishedhavum adangiyirikkunna oru sidhaanthamaanu oru samboornna sidhaantham.
oru ghadana thripthippeduthunna ellah vaakyangaludeyum samboornna sidhaanthathe "aaa gadanayude sidhaantham" ennum vilikkunnu.
oru sidhaantham sthiramaanenkil maathram athinu oru maathrkayundennathu godelinte poornnatha sidhaanthathinte (addehathinte apoornnatha sidhaanthangalumaayi thettidharikkaruthu) anantharafalamaanu.iii. eee sidhaantham vairudhyamonnum theliyikkunnilla.
athinaal, maathrukaa sidhaanthakar palappozhum "thripthikaram" ennathinu paryaayamaayi "sthiratha" upayogikkunnu.
adisthaana maathrukaa-saidhaanthika ashayangal.
oro chihnavum oru sthiramaaya chihnamo allengil oru nirddishta arittiyulla oru functiono bandha chihnamo aaya yuktirahita chinnangalude oru koottamaanu oppu allengil bhasha. chila saahityangalil, sthiramaaya chinnangal poojyam arittiyulla function chinnangalaayi kanakkaakkappedunnu, athinaal avaa ozhivaakkappedunnu. oru ghadana ennathu oru koottam soothravaakyamaanu, oppitta oro chinnangaleyum soothravaakyathile bandhangalum pravarthanangalumaayi vyaakhyaanikkunnu (oru gadanayude "vyaakhyaanam" enna aupachaarika ashayavumaayi thettidharikkaruthu). 
udaaharanathina kramappeduthiya valayangalude oru pothu oppu formula 14 aanu, ivide formula 15 umm formula 16 umm 0-aarii function chinnangalaanu (sthiramaaya chinnangal ennum ariyappedunnu), formula 17 umm formula 18 umm bainari (= 2-aarii) function chinnangalaanu, formula 19 oru yoonari (= 1-aarii) function chihnamaanu, formula 20 oru bainari relation chihnamaanu. appol, eee chinnangal formula _ 21-leeii avayude saadhaarana ardhavumaayi poruthappedunnuvennu vyaakhyaanikkumbol (angane e.jii. formula _ 22 ennathu formula _ 23 muthal formula _ 21 vareyulla oru function formula _ 20 ennathu formula _ 23) nte upavibhagavumaanu, oraalkk oru structure formula _ 27 labikkum. 
formula 28-nuu munbu vyakthamakkiya oppukalude vyaakhyaanavumaayi bandhappett formula 9-leeii oro vaakyavum formula 31-ll shariyaanenkil oru structure formula 28 nalkiyirikkunna bhaashayile phastt oordar vaakyangalude formula 29-nte oru koottam maathrkayaakkumennu parayappedunnu. (veendum, oru gadanayude "vyaakhyaanam" enna aupachaarika ashayavumaayi thettidharikkaruthu)
oru σ-structure formulayude substructure formula _ 33 athinte domaininte oru upavibhagamaanu, athinte signature σ-leeii ellah functionukalkkum keezhil adanjirikkunnu, ithu σ-leeii ellah functionukalum bandhangalum upavibhagavumaayi parimithappeduthunnathi oru σ-gadanayaayi kanakkaakkappedunnu.
ithu beejaganithathil ninnulla samana aashayangale saamaanyavalkkarikkunnu; udaaharanathinu, oru upagroop gunanavum vipareethavumaaya signecharile oru substructureanu.
eathenkilum phastt oordar formula φ-num eathenkilum moolakangalkk "a" 1, umm aanenkil oru substructurine "praadhamikam" ennu parayunnu..., soothravaakyam _ 33-nte a" "n", "
prathyekichum, "φ" oru vaakyavum formula 33 umm formula 34-nte praadhamika substructuremanenkil, formula 41 aanenkil maathram formula 40. athinaal, oru praadhamika substructure ennathu superstructure oru model aayirikkumbol krithyamaayi oru sidhaanthathinte oru modalaan. 
udaaharanam beejaganita samkhyakalude formula 42 enna feeld sankeernna samkhyakalude formula 43 enna feeldinte praadhamika uparithalamaanengilum, yukthisahamaya feeld formula 44 allah, namukk prakadippikkaan kazhiyunnathupole "2 nte oru varggamoolamundu" formula 43 kondu thripthippeduthunna aadya-krama vaakyamaayi, pakshe formula 44 kondu allah.
oru σ-structure formula _ 33 mattoru σ-structure formula _ 34-ll ulcherkkunnathu oru mapp "f": domainukalkkidayil "a" "→" b "aanu, ithu formula _ 34-nte substructureulla formula _ 33-nte isomorphisamaayi ezhuthaam. oru praadhamika substructureulla isomorphisamaayi ezhuthaan kazhiyumenkil, athine praadhamika ulcherkkal ennu vilikkunnu. oro ulcherkalum oru injective homomorphisamaanu, ennaal grouppukalilo feeldukalilo polulla bandha chinnangal oppittillengil maathrame vipareethamaayi nilanilkkoo.
oru feeld allengil vekttar space athinte gadanayude chila bhagangal avaganichukondu oru (commutative) groupaayi kanakkakkaam. oru ghadanaye yadhaartha oppinte upavibhaagathilekku "kuraykkuka" ennathaanu maathrukaa sidhaanthathile anubandha ashayam. ethir bandhathe "vipuleekaranam"-iii ennu vilikkunnu.jii. signature {+, 0} leeii oru gadanayaayi kanakkaakkappedunna yukthisahamaya samkhyakalude (additteev) groupine signature {×, +, 1,0} ulla oru feeldilekko signature {+, 0, <} ulla oru oordar cheytha groupileko vipuleekarikkaam.
athupole, σ 'mattoru signature σ vipuleekarikkunna oru signecharaanenkil, oru samboornna σ'-sidhaantham athinte vaakyangalude koottatthe σ-soothravaakyangalude koottavumaayi murichukadakkunnathiloode σ aayi parimithappedutthaam. neremarichu, oru samboornna σ-sidhaanthathe oru σ '-sidhaanthamaayi kanakkakkaam, oraalkk athu (onniladhikam vazhikalil) poornnamaaya σ'-sidhaanthathilekku vyaapippikkaan kazhiyum. reduct, expansion ennee padangal chilappol eee bandhatthilum prayogikkunnu.
compactness lovenheim-scholim sidhaanthavum.
s-nte oro parimithamaaya upavibhaagavum thripthikaramaanenkil oru koottam vaakyangal s thripthikaramaanennu compactness sidhaantham prasthaavikkunnu. thripthikaramaaya" ennathinupakaram "sthirathayulla" enna samana prasthavana nissaaramaan, kaaranam oro thelivinum thelivil parimithamaaya ennam mungaamikal maathrame upayogikkaan kazhiyoo. poornnatha sidhaantham ithu thripthikarathayilekku maattaan namme anuvadikkunnu. ennirunnaalum, compactness sidhaanthathinte niravadhi nerittulla (ardhaparamaaya) thelivukalum undu.
oru upasamhaaramaayi (i.iii.compactness sidhaantham parayunnath, thripthikaramallaatha oro phastt oordar sidhaanthathinum thripthikaramallaatha oru parimithamaaya upavibhaagamundennu. compactness" enna vaakkukal saadhaaranamaayirikkunna model sidhaanthathil eee sidhaanthathinu kendra praadhaanyamund.
phastt-oordar model sidhaanthathinte mattoru moolakkallu lovenheim-scholim sidhaanthamaanu. 
lovenheim-scholim sidhaantham anusarichu, kanakkakkavunna oppittulla oro ananthamaaya gadanaykkum kanakkakkavunna praadhamika uparithalamundu. neremarichu, eathu ananthamaaya cardinal κ-num, κ-ll thaazheyulla cardinality ulla oru ennaavunna oppile oro ananthamaaya ghadanayum cardinalityude mattoru ghadanayil adisthaanaparamaayi ulcherkkaan kazhiyum (annamillaatha oppukalkk neraaya saamaanyavalkkaranam undu). prathyekichum, lovenheim-scholim sidhaantham soochippikkunnathu ananthamaaya modalukalulla kanakkakkavunna oppilulla eathoru sidhaanthathinum kanakkakkavunna modalum ekapaksheeyamaayi valia modalukalum undennaanu.
lindstrom sidhaantham krithyamaayi paranja oru prathyeka ardhathil, lovenheim-scholim sidhaanthavum compactness sidhaanthavum nilanilkkunna ettavum expressive lojikkaanu phastt-oordar lojik.
nirnnayakatha.
nirvachikkaavunna settukal.
maathrukaa sidhaanthathil, nirvachikkaavunna settukal padanatthinte pradhana vasthukkalaan. udaaharanathinu, formula _ 51 ll formula
prime nambarukalude upavibhaagathe nirvachikkunnu, athesamayam formula
erattasamkhyakalude upavibhaagathe nirvachikkunnu.
samanamaaya reethiyil, "n" free veriyabilukalulla soothravaakyangal formula _ 54-nte upavibhaagangale nirvachikkunnu. udaaharanathinu, oru feeldil, formula
ellah soothravaakyangaludeyum vakram nirvachikkunnath aaa soothravaakyam _ 57 aanu.
ivide paraamarshichirikkunna rund nirvachanangalum "paaraameettar free" aanu, athaayath nirvachikkunna soothravaakyangal sthira domaine gadakangalonnum paraamarsikkunnilla. ennirunnaalum, "modalil ninnulla paaraamettarukal" enna nirvachanangalum oraalkk pariganikkaam.
udaaharanathinu, formula _ 58 ll, formula
oru vakram nirvachikkaan formula 58 ll ninnulla paaraameettar formula _ 60 upayogikkunnu.
quandifierukal illaathaakkunnu.
pothuve, kwaandifayarukalillaathu nirvachikkaavunna settukal vivarikkan eluppamaanu, athesamayam nested quandifierukal ulppedunna nirvachikkaavunna settukal valare sankeernamaayirikkum.
ithu nirvachikkaavunna settukal vishakalanam cheiyunnathinulla nirnaayaka upakaranamaayi quantifier elimineshane maattunnu 
oro phastt oordar formula φ ("x" 1,) aanenkil "t" enna sidhaanthathinu quantifier elemination undu..., x" "n") athinte oppinu mukalilulla modulo "t" oru phastt oordar formula σ ("x" 1,) nuu thulyamaanu..., x" n ") kwaandifayarukalillaathe, i.iii. formula _ 62 "tii" yude ellah modalukalilum nilanilkkunnu.
oru gadanayude sidhaanthathinu quantifier elemination undengil, oru ghadanayil nirvachikkaavunna oro settum yadhaartha nirvachanatthinte athe paaraamettarukalil oru quantifier free formula nirvachikkaavunnathaan.
udaaharanathinu, signature σring = (×, +, −, 0,1) leeii beejaganithaparamaayi clossed feeldukalude sidhaanthathinu quantifier elemination undu. ithinartham beejaganithaparamaayi adacha feeldil, oro soothravaakyavum polynomialukal thammilulla samavaakyangalude booliyan samyojanathinu thulyamaanu.
oru sidhaanthathinu quantifier elemination illengil, oraalkk athinte oppilekku adhika chinnangal cherkkaan kazhiyum. prathyeka sidhaanthangalkkulla, prathyekichu beejaganithathil, svayamsidhatha, alavu illaathaakkal phalangal enniva maathrukaa sidhaanthathinte aadyakaala naazhikakkallaayirunnu. ennaal palappozhum alakkunna alavu illaathaakkunnathinuparam durbalamaaya oru swath mathiyakum
tii" yude maathrkayaaya "tii" yude oru maathrkayude oro uparithalavum oru praadhamika uparithalamaanengil "tii" sidhaanthathe maathruka-poornnatha ennu vilikkunnu. oru substructure oru praadhamika substructure aano ennu parishodhikkunnathin upayogapradamaaya oru maanadandamundu, ithine tarsky-vottu testt ennu vilikkunnu. oro phastt oordar formula φ (x "1,) aanenkil maathrame oru sidhaantham" t "model poornnamaakoo ennu eee maanadandatthil ninnu pinthudarunnu..., x" "n") athinte oppinu mukalilulla modulo "t" oru asthithvaparamaaya phastt oordar formulakku thulyamaanu, i.iii. thaazhepparayunna roopathinte oru soothravaakyam
ivide si ennathu quantifier free aanu. maathruka poortthiyaakaattha oru sidhaanthathinu oru maathruka poorttheekaranamundaakam, ithu pothuvaayi, yadhaartha sidhaanthathinte vipuleekaranamallaatha oru anubandha maathruka poornna sidhaanthamaanu. oru maathrukaa pankaaliyekkurichullathaanu kooduthal pothuvaaya dhaarana.
ettavum kuranjathu.
oro ghadanayilum, oro parimithamaaya upasett formula _ 64 paaraamettarukal upayogichu nirvachikkaavunnathaan formula upayogikkuka
namukk eee soothravaakyam nishedhikkaan kazhiyunnathinaal, oro kophinaitt upavibhaagavum (domaininte parimithamaaya niravadhi gadakangalozhike ellaam ulppedunna) ellaayppozhum nirvachikkaavunnathaan.
ithu "minimum structure" enna aashayathilekku nayikkunnu. 
formula 66 ll ninnulla paaraamettarukal upayogichu nirvachikkaavunna oro upasett formula parimithamo kophinittoo aanenkil oru structure formula 66 neey minimum ennu vilikkunnu.
sidhaanthangalude thalatthilulla anubandha ashayathe "sakthamaaya minimaality" ennu vilikkunnu
tii" yude oro modalum minimum aanenkil "tii" enna sidhaanthathe "valare minimum" ennu vilikkunnu.
aaa gadanayude sidhaantham valare kurravaanengil oru ghadanaye "shakthamayi minimum" ennu vilikkunnu. thulyamaayi, oro praadhamika vipuleekaranavum kuranjathaanengil oru ghadana valare kuravaanu.
beejaganithaparamaayi adacha feeldukalude sidhaanthathinu quantifier elemination ullathinaal, beejaganithaparamaayi adacha feeldinte nirvachikkaavunna oro upavibhaagavum oru variable quantifier free formula nirvachikkaavunnathaan. oru variable quantifier free soothravaakyangal oru variable polynomial samavaakyangalude booliyan kombineshanukal prakadippikkunnu, koodaathe oru variable nontriviyal polynomial samavaakyathin parimithamaaya ennam pariharangal maathrame ulloo ennathinaal, beejaganithaparamaayi adacha feeldukalude sidhaantham valare kuravaanu.
maruvashathu, yadhaartha samkhyakalude feeld formula _ 58 athra kurravalla udaaharanathinu, nirvachikkaavunna settu pariganikkuka.
ithu parimithamo kophinittoo allatha negatteevu allatha riyal nambarukalude upavibhaagathe nirvachikkunnu.
yadhaartha samkhyaasoochikayil ekapaksheeyamaaya edavelakal nirvachikkaan oraalkk soothravaakyam _ 71 upayogikkaam.
formula _ 58-nte nirvachikkaavunna oro upavibhaagatheyum prathinideekarikkaan iva mathiyaakumenn ithu maarunnu.
kramappeduthiya gadanakalude maathrukaa sidhaanthathil minimalittiyude eee saamaanyavalkkaranam valare upayogapradamaanu.
formula 66 ll ninnulla paaraameettarukalode nirvachikkaavunna oro upasett formula pooyintukaludeyum edavelakaludeyum parimithamaaya samyojanamaanengil, oordar bandhatthinte chihnam ulppede oru oppile saandramaaya poornnamaayum kramappeduthiya ghadana formula _ 66 neey oo-minimal ennu vilikkunnu.
nirvachikkavunnathum vyaakhyaanikkaavunnathumaa gadanakal.
nirvachikkaavunna settukal prathyekichum pradhaanamaan, avaa substructureukal koodiyaanu, i. iii. ellah sthiraankangalum adangiyirikkunna iva function aaplikkeshanu keezhil adanjirikkunnu. udaaharanathinu, oru prathyeka grouppinte nirvachikkaavunna upagrooppukalekkurichu padikkaam.
ennirunnaalum, ore oppilulla substructureukalilekku svayam parimithappeduthenda aavasyamilla. n" free veriyabilukalulla soothravaakyangal formula _ 54 nte upavibhaagangale nirvachikkunnathinaal, n "-ary bandhangalum nirvachikkaavunnathaan. function graph nirvachikkaavunna oru bandhamaanengil functionukal nirvachikkaavunnathaan, koodaathe formula _ 80 sariyaanennu formula _ 66-nte eaka ghadakam "a" aanenna formula _ 78 undengil sthiraamka formula _ 77 nirvachikkaavunnathaan.
eee reethiyil, jyaamitheeya sthiratha sidhaanthathil pradhaanamaaya pothuvaaya gadanakalile nirvachikkaavunna grouppukalum feeldukalum padikkaan kazhiyum.
oraalkk oru padi koodi munnott pookaanum udanadi substructureukalkkappuram neengaanum kazhiyum.
oru ganithaghadana nalkiyaal, oru thulyatha bandhatthiloode yadhaartha gadanayude bhagamaaya oru bhaagamaayi nirmmikkaan kazhiyunna valare palappozhum bandhapetta gadanakalundu. oru pradhana udaaharanam oru grouppinte oru kwoshyant grouppaanu. 
poornna ghadana manasilakkaan oraal eee bhagangal manassilaakkanamenn oraal paranjekkaam. thulyatha bandham nirvachikkappedumbol, namukk munbathe vaakyathin krithyamaaya ardham nalkaan kazhiyum. eee gadanakal "vyaakhyaanikkaavunna"thaanennu njangal parayunnu.
vyaakhyaanikkappetta gadanakalude bhashayil ninnu vaakyangal yadhaartha gadanayude bhaashayilekku vivarthanam cheyyaan kazhiyum ennathaanu oru pradhana vastutha. athinaal oru ghadana soothravaakyam _ 66 sidhaantham anishchithathvamulla mattoraale vyaakhyaanikkukayaanenkil, soothravaakyam _ 66 thanne anishchithathvathilaanenna oraalkk kaanikkan kazhiyum.
tharangal.
adisthaana ashayangal.
oru structure formula _ 66-leeii formula _ 83-nteyum formula _ 66-nte oru upavibhagamaaya "a"-nteyum moolakangalude oru shreniyil, formula _ 83 kondu thripthippeduthunna "a"-leeii paaraamettarukalulla ellah phastt-oordar formula formulakalude koottam pariganikkaan kazhiyum. ithine "poornnamaaya (n-) tharam ennu vilikkunnu.
a" yill sthirathayulla formula _ 66-nte automorphism undengil formula _ 83-neey formula _ 91-lake ayaykkukayaanenkil, formula _ 83-umm formula _ 91-umm "a"-nuu mukalil ore poornna tharam thirichariyunnu.
oordar relations ({<}) maathram ulla oru gadanayaayi kaanunna riyal nambar line formula _ 58 eee vibhaagathil oru udaaharanamaayi varthikkum.
oro moolaka soothravaakyavum _ 95 shoonyamaaya settil ore 1-tharam thripthippeduthunnu. a", "b" ennee rund yadhaartha samkhyakale ellah samkhyakaleyum "b-a" kondu mattunna automorphism kramam bandhippichirikkunnathil ithu vyakthamaan. oru jodi samkhyakalude formula _ 96 kondu manassilaakkappedunna shoonyamaaya settinu mukalilulla poornnamaaya 2-tharam avayude kramatthe aasrayichirikkunnu formula _ 97, formula _ 98 allengil formula _ 99.
poornnasamkhyakalude 100 enna upasett soothravaakyathil, oru poornnasamkhyayallaatha yadhaartha samkhyayaaya "a" yude 1 tharam athinte moolyathe aasrayichirikkunnu.
kooduthal pothuve, formula _ 66 oru ghadanayum "a" formula _ 66-nte upavibhaagavum aakumbozhellaam, a (bhagika) "n-type ovar a" ennathu formulayude praadhamika ekstenshan formula _ 103-ll manassilakkunna paramavadhi "n" free veriyabilukalulla "p" soothravaakyangalude oru koottamaanu. 
p" ll atharam ellah soothravaakyangalum allengil athinte nishedhavum adangiyirikkunnuvenkil, "p" ennathu "poornnamaanu". a" nuu mukalilulla poornnamaaya "n"-tharangalude koottatthe palappozhum formula 105 ennu ezhuthunnu. "a" shoonyamaaya koottamaanengil, type space formula 66-nte sidhaantha formula _ 9-neey maathram aasrayichirikkunnu. formula _ 9-nuu anusrithamaayi shoonyamaaya koottatthinu mukalilulla tharangalude koottatthinu notteshan formula _ 108 saadhaaranayaayi upayogikkunnu. 
p" ennathil "113" enna oro soothravaakyathinum "112" enna soothravaakyam soochippikkunna oru soothravaakyam 110 undengil "p" neey "isolated" ennu vilikkunnu.
58 enna yadhaartha samkhyakalude soothravaakyam archemedian aayathinal, oro poornnasamkhyayekkaalum valia yadhaartha samkhyayilla. ennirunnaalum, oru compactness aargyument kaanikkunnath eathu poornnasamkhyayekkaalum valia oru moolakam ulla yadhaartha samkhya variyude praadhamika vipuleekaranamundennu.
athinaal, soothravaakyangalude koottam formula 115 ennathu oru 1-tharam ovar formula 100 aanu, athu riyal nambar line formula 58 ll saakshaathkarikkappedunni.
a" nuu mukalil oru prathyeka tharam manassilakkunna formula 54-nte gadakangalaayi krithyamaayi prakadippikkaan kazhiyunna formula 54-nte oru upavibhaagathe "a"-nuu mukalil "type-defenible" ennu vilikkunnu.
oru beejaganita udaaharanathinu, soothravaakyam _ 120 beejaganithaparamaayi adacha feeldaanennu karuthuka. eee sidhaanthathinu quantifier elemination undu. oru tharam athil adangiyirikkunna polynomial samavakyangalaal krithyamaayi nirnnayikkappedunnave kaanikkan ithu namme anuvadikkunnu. athinaal oru subfield formula 121-tharangalude poornnamaaya formulayude koottam _ 122 polynomial ring formula _ 123-nte prime idealukalude koottavumaayi poruthappedunnu, koodaathe type nirvachikkaavunna settukal krithyamaayi affine enangalaanu.
ghadanayum tharangalum.
ellah ghadanayilum ellah tharangalum thirichariyappedunnille, oro ghadanayum athinte otappetta tharangal thirichariyunnu.
oru ghadanayil thirichariyappedunna shoonyamaaya settinu mukalilulla oreyoru tharam otappetta tharangalaanenkil, ghadanaye "aattomik" ennu vilikkunnu.
maruvashathu, oro paaraameettar settilum oru ghadanayum ellah tharangalum thirichariyunnilla; oraal formula 66 muzhuvanum paaraameettar settaayi edukkukayaanengil, formula 66 ll thirichariyaan kazhiyunna oro 1-type oru formula 127-nuu "a = x" enna formula upayogichu verthirikkappedunnu. ennirunnaalum, formula 66-nte sariyaaya praadhamika vipuleekaranathil formula 66-ll "allah" enna oru ghadakam adangiyirikkunnu. athinaal oru durbalamaaya ashayam avatharippichu, athu ellah tharangalum thirichariyaan kazhiyumennu prathekshikkunna oru gadanayude ashayam pidichedukkunnu.
formula 66 nekkaal cheria cardinality ulla oru paaraameettar settu formula 130 ll ellah tharangalum manassilaakkukayaanenkil oru ghadanaye "saturated" ennu vilikkunnu.
a" yill sthirathayulla oru automorphism ellaayppozhum "a" yill tharangal samrakshikkumbol, "a" yill ore tharam thripthippeduthunna formula _ 83, formula _ 91 ennee rund sequenceukal atharam automorphism vazhi parasparam mapp cheyyaan kazhiyumennathu pothuve shariyalla. formula 66 nekkaal cheria cardinalityude ellah "eiii" kalkkum eee kanvezhs nilanilkkunna oru structure formula homogeneous ennu vilikkunnu.
yadhaartha samkhya rekha 136 enna krama soothravaakyam maathram adangiyirikkunna bhashayil aattomik aanu, kaaranam formula _ 58 leeii formula _ 83 vazhi manassilaakkappedunna shoonyamaaya settinullile ellah "n"-tharangalum formula _ 83 thammilulla krama bandhangalal verthirikkappedunnu. ennirunnaalum, ithu poorithamalla, kaaranam kanakkakkavunna settu formula _ 140 nuu mukalilulla oru tharavum ithu thirichariyunnilla, athu soochippikkunnathu "x" eathu poornnasamkhyayekkaalum valuthaanennu.
44 enna yukthisahamaya samkhyayude rekhaasoochana poorithamaanu, ithinu vipareethamaayi, formula 44 thanne kanakkakkaavunnathinaal poorithamaakaan parimithamaaya upasettukalil maathram tharangal thirichariyanam.
kallu edangal.
chila paaraamettarukal formula 122-ll nirvachikkaavunna formula 54-nte upasettukalude koottam oru booliyan beejaganithamaanu. booliyan aaljibrakalude kallinte praathinidhya sidhaantham anusarichu oru swaabhaavika eratta topological space undu, athil formula _ 122 nuu mukalilulla poornna formula _ 121-tharangal krithyamaayi adangiyirikkunnu. single formula formula _ 71-nuu _ 147 enna formula formulayude settukal srishticha topology. ithine "eiii-nuu mukalilulla enn-tharangalude stoon space" ennu vilikkunnu.
eee topology model sidhaanthathil upayogikkunna chila padangal vishadeekarikkunnu compactness sidhaantham parayunnath kallu oru compact topological space aanennum kallu topologyil "pii" oru otappetta poyintaanengil maathram oru tharam "pii" verthirikkappedumennum parayunnu.
beejaganithaparamaayi adacha feeldukalile tharangal polynomial ringinte spectravumaayi poruthappedunnunde, type spacile topology constructable topologyaanu oru koottam tharangal adisthaana ooppan aanu, athu formula _ 149 allengil formula _ 150 aanenkil. ithu sariski topologyekkaal mikachathaanu.
nirmmaana maathrkakal.
tharangal thirichariyukayum ozhivaakkukayum cheiyunnu.
chila tharangal thirichariyukayum mattullava thirichariyaathirikkum cheiyunna maathrkakal nirmmikkunnath maathrukaa sidhaanthathile oru pradhana joliyaanu.
oru tharam thirichariyaathirikkunnathi "ozhivakkal" ennu vilikkunnu, ithu saadhaaranayaayi "(kanakkakkavunna) ozhivakkal tharangalude sidhaantham" vazhi saadhyamaan
ithu soochippikkunnathu, ennaavunna oppilulla oru sidhaanthathinu shoonyamaaya settinu mukalil ennaavunna niravadhi tharangal maathrame ulloovenkil, eee sidhaanthathinu oru aattomik model undu ennaanu.
maruvashathu, oru nischitha paaraameettar settilulla eathenkilum tharam tharangal saakshaathkarikkunna oru praadhamika vipuleekaranam ellaayppozhum undu
ennirunnaalum, paaraameettar settu sthiramaayathinaal formula _ 103-nte cardinalityekkurichu ivide paraamarshamillaathathinaal, oro sidhaanthathinum oru pooritha maathrkayundennu ithu soochippikkunnilla.
vaasthavathil, oro sidhaanthathinum oru pooritha maathrkayundo ennathu settu sidhaanthathinte sermelo-frenkel sidhaanthangalil ninnu swathanthramaanu, koodaathe saamaanyavalkkaricha thudarcha sidhaantham nilanilkkunnundenkil athu sariyaanu.
ultraprodectukal.
chila tharangal thirichariyunna modalukal nirmmikkunnathinulla oru pothu saankethikavidyayaayi ultraprodectukal upayogikkunnu.
mikkavarum ellah entrikalilum yojikkunna tappilukal thirichariyaan "i" enna soochika settilulla oru koottam gadanakalude nerittulla uthppannathil ninnu oru "ultraprodect" labikkum, avide "i" ennathil oru ultrafilter "u" upayogichu "mikkavarum ellaam" krithyamaayi nirnnayikkunnu. ore gadanayude pakarppukalude ultraprodect "ultrapower" ennariyappedunnu.
maathrukaa sidhaanthathil ultraprodectukal upayogikkunnathinulla thacol "λος 's theorem" aannu
prathyekichum, oru sidhaanthathinte maathrkakal eathoru ultraprodectum aaa sidhaanthathinte oru maathrkayaanu, athinaal rund maathrkakalkku isomorphic ultrapowerukal undengil avaa moulikamaayi thulyamaanu.
keysler-shelah sidhaantham" oru vipareetharoopam nalkunnu
athinaal, aadya krama sidhaanthangal paraamarshikkunnath ozhivaakkunna praadhamika thulyathayekkurichu samsaarikkanulla oru maarggam ultraprodectukal nalkunnu. compactness sidhaantham polulla maathrukaa sidhaanthathinte adisthaana sidhaanthangalkku ultraprodectukal upayogichu badal thelivukalundu, avaa nilavilundenkil pooritha praadhamika vipuleekaranangal nirmmikkaan avaa upayogikkaam.
vargeekaranam.
isomorphism vareyulla ghadana nirnnayikkukayaanenkil oru sidhaanthathe yadhaarthathil "kattagarikkal" ennu vilichirunnu. phastt oordar lojikkinte expressivityile gurutharamaaya niyanthranangal kaaranam eee nirvachanam upayogapradamallennu maarunnu. lovenheim-scholim sidhaantham soochippikkunnathu oru sidhaanthathinu "tii" nuu chila ananthamaaya cardinal nambarinu ananthamaaya model undengil, mathiyaya valia cardinal nambarinu κ valuppathinte oru model undennaanu. vyathyastha valuppathilulla rund modalukal isomorphic aakaan kazhiyaathathinal, parimithamaaya gadanakale maathrame oru kattagarikkal sidhaanthathiloode vivarikkan kazhiyoo.
ennirunnaalum, oru kardinal κ-yude κ-kattagaritti enna durbalamaaya ashayam model sidhaanthathile oru pradhana aashayamaayi mariyirikkunnu. tii" yude eathenkilum rund maathrkakal isomorphic aanenkil "tii" enna sidhaanthathe "κ-kattagarikkal" ennu vilikkunnu. κ-vargeekaranathekkurichu chodyam bhaashayude kardinaalithyathekkaal valuthaano ennathine vimarsanaathmakamaayi aashrayichirikkunnuvennu maarunnu (i.iii. soothravaakyam _ 175 + |σ|, evide |σ| oppukalude kardinaalthathayaanu). parimithamo kanakkakkaavunnatho aaya oppukalkk, athaayath, 176-cardinality formula κ-cardinality formula thammil adisthaanaparamaaya vyathyaasamund.
ω-vargeekaranam.
soothravaakyam _ 176-kattagarikkal sidhaanthangale avayude tharam sthalathinte gunangalaal savisheshamaakkaam
formula _ 180 sidhaantham koodiyaaya formula _ 179 sidhaantham formula _ 176-kaattagarikkalaan, kaaranam shoonyamaaya settinu mukalilulla oro "n"-type formula _ 182 umm formula _ 183 thammilulla jody krama bandhathaal verthirikkappedunnu.
ithinartham kanakkakkavunna oro saandramaaya rekheeya kramavum yukthisahamaya samkhyaasankethathinu kramam-isomorphic aanu. maruvashathu, formula _ 44, formula _ 58, formula _ 43 ennivayude sidhaanthangal formula _ 176-kattagarikkal allah. eee feeldukalilellaam, ananthamaaya niravadhi swaabhaavika samkhyakalil eathenkilum oru formula 188 kondu nirvachikkaan kazhiyum enna vasthuthayil ninnaan ithu pinthudarunnathu.
formula _ 175-kattagarikkal sidhaanthangalkkum avayude kanakkakkavunna modalukalkkum oligomorphic grouppukalumaayi sakthamaaya bandhamunda
enchelar, rillu-nardsevsky, svenonius enniva swathanthramaayi eee upavibhaagathinte thulyamaaya swabhaavasavisheshathakale chilappol rillu-nardsevsky sidhaantham ennu vilikkunnu.
samyojitha oppukalil, soothravaakyam _ 176-kattagarikkal sidhaanthangalude oru pothu uravidam frashe paridhikalaan, avaa parimithamaaya bandhita gadanakalude oru klaasinte saadhyamaaya ellah konfigareshanukalum samyojippikkunnathinulla paridhiyayi labikkum.
ennaanaavaatha vargeekaranam.
kanakkakkavunna bhaashakalile sidhaanthangalkku "ennaanaavaatha vargeekaranam" enna oru ashayam maathrameyulloo ennu 1963 ll maikkal morley kaanichu.
morleyude thelivukal kanakkukoottaanaavaatha vargeekaranavum modalukalude aantharika ghadanayum thammilulla aazhathilulla bandhangal velippeduthi, ithu vargeekarana sidhaanthathinteyum sthiratha sidhaanthathinteyum prarambha pooyintaayi maari. 
ennamatta vargeeka sidhaanthangalaanu pala kaazhchappaadukalil ninnum ettavum nannayi perumaarunna sidhaanthangal.
prathyekichum, poornnamaaya sakthamaaya minimum sidhaanthangal ennamatta varggeekrithamaanu. oru prathyeka swabhaavathinte beejaganithaparamaayi adacha feeldukalude sidhaantham ennamatta varnnanaparamaanennum feeldinte athirukadanna digri athinte isomorphism tharam nirnnayikkunnuvennum ithu kaanikkunnu.
176-kattagarikkal, ennamatta kattagarikkal ennee soothravaakyangalulla oru sidhaanthathe "poornnamaayum kattagarikkal" ennu vilikkunnu.
sthiratha sidhaantham.
oru phastt-oordar sidhaanthathinte modalukalude klaas gadanayile oru pradhana ghadakam "sthiratha shreniyil" athinte sthaanamaanu. 
oru sidhaantham ananthamaaya cardinal formula 193-nuu formula 193-sthiramaanenkil athine "sthiratha" ennu vilikkunnu. parambaraagathamaayi, formula 175-sthirathayulla sidhaanthangale "formula 176-sthiratha" ennu vilikkunnu.
sthirathayude shreni.
sthiratha sidhaanthathinte adisthaana phalam "sthiratha spectrum sidhaantham" aanu, ithu soochippikkunnathu oru ennaavunna oppilulla oro samboornna sidhaanthavum "tii" inipparayunna klaasukalilonnil varunnuvennu
aadya tharam sidhaanthathe "asthira" ennum randaamathe tharam sidhaanthathe "karsanamaayi sthirathayulla" ennum moonnaamathe tharam sidhaanthathe "superstable" ennum vilikkunnu.
koodaathe, oru sidhaantham formula 176-sthiramaanenkil, athu ellah ananthamaaya kardinaalukalilum sthirathayullathaanu, athinaal formula 176-sthiratha andhavishwaasathekkaal shakthamaan.
sthiramaaya sidhaanthangalil parimithappedutthumbol maathrukaa sidhaanthathile pala nirmmaanangalum eluppamaanu; udaaharanathinu, saamaanyavalkkaricha thudarcha sidhaantham shariyaano ennathu pariganikkathe, oru sthiramaaya sidhaanthathinte oro modalinum oru pooritha praadhamika vipuleekaranamundu.
sthirathayulla sidhaanthangal padikkunnathinulla shellayude yadhaartha prachodanam oru kanakkakkavunna sidhaanthathinu ethra maathrkakal undennu theerumaanikkuka ennathaayirunnu. 
oru sidhaantham kanakkillaatha tharamthirikkaavunnathaanenkil, athu 176-sthirathayulla soothravaakyamaanu. kooduthal pothuve, "pradhana vidavu sidhaantham" soochippikkunnathu, oru sidhaanthathinu "t" ennathinu "cardinality formula 193" enna formulayude 215 modalukalekkaal kurravulla oru annamillaatha cardinal formula 193 undengil, "t" superstable aanu.
jyaamitheeya sthiratha sidhaantham.
oru sidhaanthathinte maathrkaykkullil nirvachikkaavunna settukalude jyamiti vishakalanam cheiyunnathinum sthirathayude shreni nirnaayakamaan. 
176-sthirathayulla sidhaanthangalude soothravaakyathil, oru modalinullile nirvachikkaavunna settukalkkulla oru pradhana alavukolaanu "morley rank". traansfainit induction aanu ithine nirvachichirikkunnathu
nirvachikkaavunna oro settinum nannayi nirvachikkapetta morley rank ulla oru sidhaanthathe "tii" ennu vilikkunnu; "tii" kanakkaakkaavunnathaanenkil, "tii" formula _ 176-sthiramaanenkil maathram "tii" poornnamaayum atheethamaayi kanakkakkum.
oru tharatthilulla morley rank aaa tharatthilulla soothravaakyangalude morley raankukalil ettavum kuranjathaayi krameekarichukondu morley rank tharangalilekku vyaapippikkaan kazhiyum. athinaal, "a" enna gadakathinte morley raankine "a" enna paaraameettar settinu mukalilulla "a" enna moolakatthinte morley rankinekkurichum samsaarikkam. 
oru sidhaantham avishwasaneeyamo (yuu-rank) allengil kevalam sthiramo (shellayude formula _ 219-rank) aanenkil maathram nannayi nirvachikkappedunna morley raankinte analogukalum undu.
swaathanthryatheyum pothuvaaya vipuleekaranangaleyum kurichulla ashayangal nirvachikkaan aaa alavukal upayogikkaam.
aduthide, sthiratha laalithyamaayi vighadippikkappettu, "swaathanthrya swathalla" (nip). 
swathanthrathayekkurichu nannayi perumaarunna oru ashayam nirvachikkaan kazhiyunna sidhaanthangalaanu lalithamaaya sidhaanthangal, athesamayam nipp sidhaanthangal oo-minimal gadanakale saamaanyavalkkarikkunnu.
oru sidhaantham nipum lalithavum aanenkil maathram sthirathayullathinaal avaa sthirathayumaayi bandhappettirikkunnu, koodaathe sthiratha sidhaanthathinte vividha vashangal eee klaasukalilonnil sidhaanthangalilekku saamaanyavalkkarikkattittu.
noonn-elementary model sidhaantham.
maathrukaa-saidhaanthika phalangal praadhamika klaasukalkkappuram, athaayath phastt-oordar sidhaantham vazhi svayamsidhamaakkaavunna classukalkku appurathekku saamaanyavalkkarikkattittu.
eee yukthikale poornnathayum othukkavum pothuve baadikkilla enna vastutha uyarnna kramatthilulla yukthikalilo ananthamaaya yukthikalilo ulla maathrukaa sidhaanthathe thadassappeduthunnu. lindstrom sidhaantham ithu urappichuparayunnu, lovenheim-scholim sidhaanthangalum compactnus sidhaanthangalum nilanirthunna ettavum sakthamaaya yukthiyaan phastt-oordar lojik ennu ekadesham prasthaavikkunnu. ennirunnaalum, eee yukthikalkkum maathrukaa saidhaanthika saankethikavidyakal vipulamaayi vikasippichittund. ennirunnaalum, kooduthal avishkaara yukthisahamaya bhaashakalude maathrukaa sidhaanthathinte bhooribhaagavum sermelo-frankel settu sidhaanthathil ninnu swathanthramaanennu thelinjirikkunnu.
aduthide, poornnamaaya sthirathayullathum vargeekarikkappettathumaaya sidhaanthangalilekkulla shradha kendreekarikkunnathinnom, oru logical sidhaantham svayamsidhamaakkunnathine ardhaparamaayi nirvachikkapetta modalukalude klaasukalil pravartthanangal nadannittundu.
ekapaksheeyamaayi valia ekathaanamaaya modalukalude substructureukalude klaas padikkunna "homogeneous model sidhaantham" oru udaaharanamaanu. sthiratha sidhaanthathinteyum jyaamitheeya sthiratha sidhaanthathinteyum adisthaana phalangal eee krameekaranatthe saamaanyavalkkarikkunnu.
sakthamaaya minimal sidhaanthangalude saamaanyavalkkaranamenna nilayil, nirvachikkaavunna oro settum kanakkakkaavunnatho qoo-countable aayato aaya klaasukalaanu kwaasiminimali bestt klaasukal. sankeernamaaya exponential functionte maathrukaa sidhaanthathinte thacolaanu avaa.
sthiratha padikkunna ettavum pothuvaaya ardhaghadana amoorthamaaya praadhamika klaasukalaanu, avaye nirvachikkunnath oru praadhamika uparithalathinte saamaanyavalkkarikkunna "sakthamaaya uparithala" bandhamaan. athinte nirvachanam poornnamaayum ardhavathaanengilum, oro amoortha praadhamika classes chila tharangal ozhivaakkunna oru phastt-oordar sidhaanthathinte modalukalaayi avatharippikkan kazhiyum. amoorthamaaya praadhamika klaasukalilekkulla sthiratha-saidhaanthika ashayangal saamaanyavalkkarikkunnath oru gaveshana paripaadiyaanu.
thiranjeduttha aaplikkeshanukal.
yadhaartha clossed feeldukal, booliyan aaljibrakal, thannirikkunna swabhaavasavisheshathakalude beejaganithaparamaayi clossed feeldukal thudangiya beejaganithaparamaayi rasakaramaaya vividha klaasukalkkulla quantifier elemination sambandhicha tarskyude thelivukalaanu model sidhaanthathinte aadyakaala vijayangalil onnu. yadhaartha-adacha, beejaganithaparamaayi adacha feeldukalude aadya-krama sidhaanthangalum booliyan aaljibrakalude aadya-krama sidhaanthavum nirnnaayakamaanennu kaanikkanum booliyan aaljibrakale praadhamika thulyathayilekku tharamthirikkaanum riyal-clossed feeldukaludeyum thannirikkunna swabhaavathinte beejaganithaparamaayi adacha feeldukaludeyum sidhaanthangal savisheshamaanennu kaanikkanum quantifier elemination tarsky anuvadichu. koodaathe, quantifier elemination beejaganithaparamaayi adacha feeldukalile beejaganita inangalaayum yadhaartha-adacha feeldukalile nirvachikkaavunna bandhangale semiologibric settukalaayum nirvachikkaavunna bandhangalekkurichu krithyamaaya vivaranam nalki. 
1960kalil ultraprodect nirmmaanatthinte aamukham beejaganithathil puthiya prayogangalilekku nayichu. parimithamaaya mandalangalude sidhaantham nirnnaayakamaanennu theliyikkunna syoodophainaitt feeldukalile aaksinte pravartthanavum diophantine samavakyangalekkurichulla aarttinte anumaanathinte prathyeka kesai axum kochanum nalkunna thelivum, axa-kochan sidhaanthavum ithil ulppedunnu. ultraprodect nirmmaanam ebraham robinsante staanderd allatha vishakalanathinte vikasanathilekku nayichu, ithu ananthamaaya alavukalude karsanamaaya calculus nalkaan lakshyamidunnu.
aduthide, sthirathayum nirvachikkaavunna settukalude jyamitiyum thammilulla bandham beejaganita, diophantine jyamithiyil ninnulla niravadhi prayogangalilekku nayichu, 2001 leeii ellah savisheshathakalilum jyaamitheeya mordell-laang anumaanathinte 1996 leeii ehud hrushovsky thelivu ulppede, maanin-mumford anumaanathinte saamaanyavalkkaranam theliyikkaan samanamaaya reethikal upayogichu.
2011-ll, modular karvukalude uthppannangalkkayi andrey-oorttu anumaanam theliyikkaan jonathan pila oo-minimalitye chuttipattiyulla saankethikavidyakal prayogichu.
sthirathayulla sidhaanthangale chuttipattiyulla oru prathyeka anweshanangalil, nipp sidhaanthangal mesheen laining sidhaanthathil paaku-lanable aaya nirvachikkaavunna klaasukale krithyamaayi vivarikkunnuvennu 1992 ll laskovsky kaanichu. ithu eee vyathyastha mekhalakal thammilulla niravadhi edapedalukalkku kaaranamaayi. 2018-ll, vettakkaaranum vettakkaaranum sthiramaaya sidhaanthangal oonlain padikkaavunna klaasukalumaayi poruthappedunnuvennu kaanikkunnathinal kathidapaadukal vipuleekarichu.
charithram.
irupathaam noottaandinte madhyam muthal oru vishayamaayi maathrukaa sidhaantham nilanilkkunnu, 1954 ll lvo-varso skoolile amgamaaya alfred tarsky aanu eee peru srishtichathu. ennirunnaalum munbathe chila gaveshanangal, prathyekichu ganithashaasthra yukthiyil, palappozhum oru maathrukaa-saidhaanthika swabhaavamullathaayi kanakkaakkappedunnu. 1915ll leopold lovenheim prasidheekaricha thaazhekkulla lovenheim-scholim sidhaanthathinte oru prathyeka kesaayirunnu ippol model sidhaanthathinte aadya pradhana phalam. compactness sidhaantham thoralf scholaminte pravartthanatthil antharleenamaayirunnu, ennaal ithu 1930 ll kurt godelinte poornnatha sidhaanthathinte thelivile lemmayayi aadyamaayi prasidheekarichu. lovenheim-scholem sidhaanthathinum compactness sidhaanthathinum 1936lum 1941lum anatoli maltsavil ninnu athath pothuvaaya roopangal labhichu.
oru swatantra vibhaagamenna nilayil maathrukaa sidhaanthathinte vikasanam konduvannathu interbellum kaalaghattatthil alfred tarskyaanu. mattu vishayangalkoppam yukthisahamaya anantharafalam, deductive sisttangal, lojikkinte beejaganitam, nirvachana sidhaantham, sathyathinte ardha nirvachanam enniva tarskyude kruthikalil ulppedunnu. addehathinte ardhashaasthra reethikal 1950kalilum 60kalilum adhehavum addehathinte niravadhi berkley vidyaarthikalum vikasippichedutha maathrukaa sidhaanthathil paryavasaanichu.
achadakkathinte thudar charithratthil, vyathyastha izhakal uyarnnuvaraan thudangi, vishayathinte shradha maari. 1960kalil, ultraproductukale chuttipattiyulla saankethikavidyakal model sidhaanthathil oru janapriya upakaranamaayi maari. athe samayam, jaims aaksineppolulla gaveshakar vividha beejaganita klaasukaludeyum echu polulla mattullavayudeyum phastt-oordar model sidhaantham anweshikkukayaayirunnu. phastt-oordar model sidhaanthathinte ashayangalum phalangalum mattu logical systengalilekku jerom keysler vyaapippikkukayaayirunu. thudarnnu, morleyude prashnatthil ninnu prachodanam ulkkond, shella sthiratha sidhaantham vikasippichu. sthirathaye chuttipattiyulla addehathinte pravartthanangal maathrukaa sidhaanthathinte roopathe maattimarichu, ithu oru puthiya klaas aashayangalkk kaaranamaayi. ithine maathrukaaparamaaya maattam ennu vilikkunnu. 
adutha dashakangalil, thathfalamaayundaakunna sthiratha shreni aaa modalukalil nirvachikkaavunna settukalude jyamithiyumaayi adutha bandhamundennu vyakthamaayi; ithu ippol jyaamitheeya sthiratha sidhaantham ennariyappedunna upavibhaagathin kaaranamaayi. jyaamitheeya maathruka sidhaanthathil ninnulla swaadheenamulla oru thelivinte oru udaaharanam function feeldukalkkaayulla mordell-laang anumaanathinte hrushovsky thelivaanu.
ganithashaasthra yukthiyude anubandha shaakhakalumaayulla bandhangal.
parimithamaaya maathruka sidhaantham.
parimithamaaya gadanakalil shradha kendreekarikkunna parimithamaaya maathrukaa sidhaantham, padicha prashnangalilum upayogicha saankethikavidyakalilum ananthamaaya gadanakalekkurichulla padanatthil ninnu ganyamaayi vyathichalikkunnu. prathyekichum, parimithamaaya gadanakalil parimithappedutthumbol paraajayappedunna classical model sidhaanthathinte niravadhi kendra phalangal. compactness sidhaantham, godelinte poornnatha sidhaantham, phastt-oordar lojikkinulla ultraprodectukalude reethi enniva ithil ulppedunnu. parimithavum ananthavumaaya maathrukaa sidhaanthathinte interfacil algoritam allengil computable maathrukaa sidhaanthavum 0-1 niyamangalude padanavum undu, avide oru klaas gadanakalude pothuvaaya sidhaanthathinte ananthamaaya maathrkakal parimitha maathrkakal vitharanam cheiyunnathinekkurichu vivarangal nalkunnu. 
vivaranathmaka sankeernnatha sidhaantham, daattaabes sidhaantham, aupachaarika bhasha sidhaantham ennivayaanu efmt pradhana prayoga mekhalakal.
sidhaantham sajjamaakkuka.
eathoru settu sidhaanthathinum (kanakkakkavunna bhashayil prakadippikkunna), athu sthiramaanenkil, kanakkakkavunna oru maathrkayundu; ithu scholaminte paradox ennariyappedunnu, kaaranam settu sidhaanthathil ennaanaavaatha settukalude nilanilpp anumaanikkunna vaakyangalund, ennittum eee vaakyangal nammude kanakkakkavunna maathrkayil sariyaanu. prathyekichum thudarchayaaya sidhaanthathinte swaathanthryathinte thelivinaayi, maathrkayil ninnu "ullil" nokkumbol ennaanaavaathathaayi thonnunna, ennaal maathrkaykku "purathulla" oraalkk kanakkakkavunna modalukalile settukal pariganikkendathundu.
maathrukaa-saidhaanthika veekshanam settu sidhaanthathil upayogapradamaanu; udaaharanathinu, nirmmitha prapanchathekkurichulla kurt godelinte pravartthanatthil, pooll kohen vikasippichedutha fozhsu cheiyunna reethiyodoppam, thiranjeduppinte svayamsidhathayude (veendum thathwashaasthraparamaayi rasakaramaaya) swaathanthryavum settu sidhaanthathinte mattu svayamsidhaanthangalil ninnulla thudarchayaaya sidhaanthavum theliyikkaan kazhiyum.
maruvashathu, sermelo-frenkel settu sidhaanthathinullil model sidhaantham thanne aupachaarikamaakkappedunnu. udaaharanathinu, maathrukaa sidhaanthathinte adisthaanaghadakangalude (compactness sidhaantham polullava) vikasanam thiranjedukkunnathinte sidhaanthathe aasrayichirikkunnu, vaasthavathil booliyan prime ideal sidhaanthathinu thiranjedukkaathe sermelo-frenkel settu sidhaanthathinu thulyamaanu. maathrukaa sidhaanthathile mattu phalangal staanderd zfc chattakkoodinapuramulla settu-theoritic soothrangale aasrayichirikkunnu. udaaharanathinu, thudarchayude sidhaantham nilanilkkunnundenkil, kanakkakkavunna oro modalinum poorithamaaya (athintethaaya kardinalittiyil) oru ultrapower undu. athupole, saamaanyavalkkaricha thudarcha sidhaantham nilanilkkunnundenkil oro modalinum oru pooritha praadhamika vipuleekaranamundu. eee phalangalonnum ZFC-yill maathram theliyikkanaavilla. avasaanamaayi, maathrukaa sidhaanthathil ninnu uyarnnuvarunna chila chodyangal (ananthamaaya yukthikalude compactness polullava) valia cardinal svayamsidhaanthangalkku thulyamaanennu theliyikkappettittund.