
oru sidhaanthathinte spectrum

ganithashaasthra yukthiyude oru shaakhayaaya maathrukaa sidhaanthathil, oru sidhaanthathinte spectrum
vividha kaardinaalittikalilulla isomorphism klaasukalude modalukalude ennamaanu ithu nalkunnathu. kooduthal krithyamaayi paranjaal, 
oru bhaashayile eathenkilum samboornna sidhaanthamaaya "tii" kku "κ" enna cardinalityude "tii" (isomorphism vare) modalukalude ennathinu "aii" ("tii", "κ") ennu ezhuthunnu. i" ("t", "κ")-nte saadhyamaaya perumaattangale "κ"-yude oru pravartthanamaayi vivarikkuka ennathaanu spectrum prashnam. tii" enna kanakkukoottaavunna sidhaanthathinte kaaryatthil ithu eathaandu poornnamaayum pariharichirikkunnu.
aadyakaala phalangal.
eee vibhaagathil "tii" ennathu kanakkakkavunna oru samboornna sidhaanthavum "κ" oru kardinaalavumaanu.
lovenheim-scholim sidhaantham kaanikkunnath oru ananthamaaya kardinaalinu "i" ("t", "κ") poojyamallengil athu ellaavarkkum poojyamalla ennaanu.
spectrum prashnam pariharikkunnathinulla aadya pradhana ghattamaayirunnu morleyude kaattagoritti sidhaantham "i" ("t", "κ") ennathu chila annamillaatha "κ" nuu 1 aanenkil athu annamillaatha ellah "κ" num 1 aanennu athu prasthaavikkunnu. 
i" ("t", "α0) 2 aakaan kazhiyillennu roobert vottu kaanichu. 2 allatha eathenkilum negatteevu allatha poornnasamkhyayulla udaaharanagal eluppathil kandethaam." i "(" t ", α0) ananthamaanengil athu α0 allengil α1 allengil 2a0 aayirikkanamenn morley theliyichu. thudarchayaaya sidhaantham thettaanengil athu α1 aano ennu ariyilla ithine vottu conjecture ennu vilikkunnu, ithu spektrathinte sidhaanthathil (2005 ll) avasheshikkunna pradhana thuranna prasnamaanu.
maikkal dii aadyamaayi nirddeshicha oru anumanamaayirunnu (ippol oru sidhaantham) morleyude prashnam. i" ("t", "κ") ennathu "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" ennathil "κ" "ennathil" κ "" ennathil "κ" "ennathil" "" "" "ennu" "" ennu ""  saharon shelhayan ithu theliyichathu. ithinaayi addeham valare aazhathilulla oru dvibhajana sidhaantham theliyichu.
spectrum prashnathinu mikkavarum samboornna pariharam saharon shelha nalki. oru samboornna sidhaanthamaaya "tii" kku, onnukil "aii" ("tii", "κ") = 2 "κ" ellah annamillaatha kardinaalukalkkum "κ", allengil ellah oordinalukalkkum formula _ 1 (notteshante vishadeekaranathinaayi alef number beth number kaanuka), ithu saadhaaranayaayi aadya kesil bandhithathekkaal valare cheruthaanu. ekadesham paranjaal onnukil annamillaatha ellah cardinalitykalilum saadhyamaaya paramavadhi modalukal undenno allengil annamillaatha ellah cardinalitykalilum "kurachu" modalukal maathrame ulluvennum ithinartham. kurachu modalukal ullappol kesil saadhyamaaya spectrayekkurichum shela oru vivaranam nalki.
kanakkakkavunna oru sidhaanthathinte saadhyamaaya spectrayude pattika.
shalhayude srishtikal vipuleekarichukondu brad hart, ehud hrashovsky, maikkal sii. ennaanaavaatha cardinalitykalile kanakkakkavunna sidhaanthangalkkaayi spectrum prashnathinu laskovsky inipparayunna samboornna pariharam nalki. 
t" ennathu kanakkakkavunna oru samboornna sidhaanthamaanenkil, isomorphism klaasukalude modalukalude nambar i ("t", αα) oordinalukalkku α> 0 nuu kuranjathu 2aα umm inipparayunna maappukalilonn nalkunnu
koodaathe, mukalil paranja ellah saadhyathakalum kanakkakkavunna chila samboornna sidhaanthathinte spectramaayi sambhavikkunnu.
mukalilulla listtile "dii" enna samkhya sidhaanthathinte aazhamaan.
t" oru sidhaanthamaanenkil, oru puthiya sidhaantham 2 "t" neey oru thulyatha bandhamullah sidhaanthamaayi nirvachikkunnu, angane ananthamaaya niravadhi thulyatha klaasukal undu, avaa oronnum "t" yude maathrkayaanu. soothravaakyam _ 14, soothravaakyam _ 15, soothravaakyam _ 16 ennivayum nammal nirvachikkunnu. 
formula _ 17. d" yude minimum moolyangalkku udaaharanangalil ninnu "d" yude minimum moolyangalkku mukalilulla pattikayil spectra ulla sidhaanthangalude udaaharanagal nirmmikkaan ithu upayogikkaam.