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I'm almost there with Godel's proof. One last thing I can't quite get my head around - how substitution can come up with a self-referential statement.

The 'plain english' version of the substitution operation can be found here, but it's still a little mystifying, so I tried a simple exercise to see if I could come up with a numbering system that could refer to itself using the substitution process described in the article, and that I can see with my own eyes.

I have a variable x, let's say it has Godel number 1.

I have a predicate B, Godel number 2 which means, let's say, "has no proof"

So xB ("x has no proof") has the G number 18 (2^1 * 3^2).

Now, as I understand it, I substitute 18 in for x.

Writing out SSSSS....SSS0 to represent 18 per the formal system would get into larger numbers than i want, so I'm going to denote the character 1 to have Godel number 3, and the character 8 to have Godel number 4.

So xB becomes 18B which has the G number 16200 (2^3 * 3^4 * 5^2)

My question is - now what? I'm no closer to self referentiality. If I substitute 16200 in for anything in that formula I'm just going to end up with an ever larger G number.

Is my simplistic numbering system wrong? Or have I misunderstood the substitution concepts? I don't follow how you can have two rounds of substitution, but end up with a constant G number, which is the basis for self referentiality.

Mauro ALLEGRANZA
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5imon
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1 Answers1

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See Diagonalization.

Self-referentiality is not so obvious to be read. We prove the general result that:

for every formula $A(x)$ we can produce a sentence $D$ such that [the system $F$ proves that] $D ↔ A(\ulcorner D \urcorner)$.

Then we apply it to the predicate $¬\text {Prov}_F(x)$ to get that system $F$ proves $G ↔ ¬\text {Prov}_F(\ulcorner G \urcorner)$.

The machinery of substitution is needed in the proof of the general result above, called Diagonal Lemma.

Mauro ALLEGRANZA
  • 99,247