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.
Ukungeniswa kwamanani eGödel (imephu yefomula engacacanga kumanani endalo) kunye nokwahlula-hlula (ukutshintshwa kwesitshintshi simahla kwimisebenzi kunye nenombolo yabo ye-Gödel) zizinto ezimbini eziphambili ezaziswa nguGödel kubungqina bakhe. Umbono wobungqina obuchazayo apho iGödel idibanisa khona la magama angabhalwa 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 \(d\) .
Ukuqhubeka kokufunda sicebisa ukupapashwa kukaGödel ngo-1931 kunye nenqaku elifundwe kakuhle nguStepan Parunashvili . Ukongeza kwiithiyori zokungagqitywa, uGödel wenze ezinye izinto eziphumelelayo, kubandakanya ukungabinakuphikiswa kweCantor's continuum hypothesis kunye nobungqina be - ontological kaThixo kulwimi lwendlela yokucinga.