Xa uKurt Gödel wapapasha iTheorems yakhe edumileyo yokungazaliseki ngo-1931, yagungqisa iziseko zengqondo yemathematika: Waphikisa ukuba zonke ii-axioms ezinokusekwa njengesiseko ezinokwenzeka ngokuqinisekileyo azigqibekanga ukuze zingqine zonke iingxelo ngamanani-kwaye wayitshabalalisa loo nto Iphupha likaHilbert lokungqina ukungqinelana kwethiyori yezibalo.
Ukwaziswa kwamanani eGödel (umzobo okhethekileyo weefomyula kumanani endalo) kunye nokudibanisa (ukutshintshwa kwe-free variable kwimisebenzi kunye nenombolo yayo yeGödel) ziingcamango ezimbini eziphambili uGödel azizisayo kubungqina bakhe. Ingcamango ebalulekileyo yobungqina apho uGödel edibanisa ezi ngcamango ingabhalwa ngolu hlobo lulandelayo.:
$$P(p) \, \text{wahr} \Leftrightarrow p \in \, \overline{B}^* \Leftrightarrow d(p) \in \overline{B} \Leftrightarrow d(p) \notin B \Leftrightarrow g(P(p)) \notin B \Leftrightarrow P(p) \, \text{unbeweisbar}$$
Kuba \(P(p)\) ayinakuba yinyani (kuba ibinokuchaphazeleka kwaye ke iyinyani), \(P(p)\) yinyani kwaye ke ayizukubakho. Ke kuhlala kukho isigwebo esiyinyani kulwimi (kunye naluphi na ukhetho lwee-axioms) ezingenakubonakaliswa. Apha \(g\) i Gödelization, \(p\) i Gödel inani isivisa \(P\) , nto leyo archetype edibanayo \(\overline{B}^*\) of \(B\) (i isethi yonke Amanani e-Godel azo zonke izindululo ezinokubakho) phantsi komsebenzi wokudibanisa \(d\) .
Ukuze ufunde okungakumbi , kucetyiswa ushicilelo lukaGödel luka-1931 kunye nenqaku likaStepan Parunashvili elinolwazi. Ngaphandle kweengcamango zokungapheleli, uGödel wenza eminye iminikelo ephawulekayo, kuquka ukungaphikiswa kwengcamango kaCantor eqhubekayo kunye nengxoxo ye-ontological yokuba uThixo ukho kulwimi lwe-modal logic.