# Umsebenzi wobuciko kaGodel1220

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.

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