Ngenkathi uKurt Gödel eshicilela iThe Incompleteness Theorems yakhe edumile ngo-1931, yazamazamisa izisekelo zomqondo wezibalo: Waphika ukuthi zonke izifinyezo ezingasethwa njengesisekelo kungenzeka ukuthi aziphelele ukuze kufakazelwe zonke izitatimende ngezinombolo - futhi wakushabalalisa lokho Iphupho likaHilbert lokufakazela ukuvumelana kombono wezibalo.
Ukwethulwa kwezinombolo ze-Gödel (imephu engaqondakali yamafomula ezinombolweni zemvelo) kanye nokuhlukaniswa (ukushintshwa kokuguquguquka kwamahhala kwemisebenzi nenombolo yabo ye-Gödel) imiqondo emibili emaphakathi eyethulwa nguGödel ebufakazini bakhe. Ubufakazi obunqumayo lapho i-Gödel ihlanganisa khona lemiqondo bungabhalwa phansi ngokulandelayo:
$$P(p) \, \text{wahr} \Leftrightarrow p \in \, \overline{B}^* \Leftrightarrow d(p) \in \overline{B} \Leftrightarrow d(p) \notin B \Leftrightarrow g(P(p)) \notin B \Leftrightarrow P(p) \, \text{unbeweisbar}$$
Njengoba \(P(p)\) kungeke kube ngamanga (ngoba kungenjalo kungacasulwa futhi kube yiqiniso), \(P(p)\) yiqiniso ngakho-ke kungabi okungahlanjululwa. Ngakho-ke kuhlale kunomusho weqiniso olimini (nganoma yikuphi ukukhetha kwama-axioms) ongenakufakazelwa. Lapha \(g\) -Gödelization, \(p\) inombolo yeGödel yesilandiso \(P\) , okuyi-archetype ehambisanayo \(\overline{B}^*\) ye \(B\) (iqoqo lawo wonke Izinombolo ze-Godel zazo zonke iziphakamiso ezingafakazelwa) ngaphansi komsebenzi we-diagonal \(d\) .
Ukuqhubeka nokufunda sincoma ukushicilelwa kukaGödel kusuka ngo-1931 kanye nendatshana ebhalwe nguStepan Parunashvili, okufanele ukukufunda . Ngaphezu kwemibono yokungapheleli, uGödel wenze ezinye izimpumelelo ezinkulu, kufaka phakathi ukungaphikeki komqondo wokuqhubeka kukaCantor kanye nobufakazi bokuthi uNkulunkulu ukhona ngolimi lwe-logic modal.