Step 1: Encode sentences in PA as numbers (regardless of if they have a free variable or not). Step 2: Encode proofs in PA as numbers. 2/4
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Step 3: If x encodes a sentence F with a free variable and y is a number, let sub(x,y) be the number encoding F(y) (i.e. y substituted into the free variable of F). Step 4: Let P(x) be the sentence "the sentence encoded by x (has no free variables and) has no proof." 3/4
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Step 5: Let n encode the sentence (with a free variable) P(sub(x,x)). Then n is simply a number. Step 6: Note that sub(n,n) encodes P(sub(n,n)). Step 7: P(sub(n,n)) is true but unprovable. 4/4
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