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 (ukumapha okuhlukile kwamafomula ezinombolweni zemvelo) kanye nokwehlukanisa (ukufakwa esikhundleni kwe-free variable emisebenzini nenombolo yayo ye-Gödel) kuyimiqondo emibili eyinhloko uGödel ayivezayo ebufakazini bakhe. Umqondo obalulekile wobufakazi lapho uGödel ehlanganisa khona le miqondo ungabhalwa kanje.:
$$P(p) \, \text{wahr} \Leftrightarrow p \in \, \overline{B}^* \Leftrightarrow d(p) \in \overline{B} \Leftrightarrow d(p) \notin B \Leftrightarrow g(P(p)) \notin B \Leftrightarrow P(p) \, \text{unbeweisbar}$$
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 \(d\) .
Ukuze ufunde kabanzi , kunconywa incwadi kaGödel yango-1931 kanye nesihloko sikaStepan Parunashvili esinokuqonda. Ngaphandle kwemibono yokungapheleli, uGödel wenza eminye iminikelo ephawulekayo, okuhlanganisa ukungabi nakuphikiswa kwenkolelo-mbono kaCantor eqhubekayo kanye nempikiswano ye-ontological yokuba khona kukaNkulunkulu ngolimi lwe-modal logic.