Gödel's incompleteness theorem - question about self reference

Question

Gödel first incompleteness theorem states that certain formal systems cannot be both consistent and complete at the same time. One could think this is easy to prove, by giving an example of a self-referential statement, for instance: "I am not provable". But the original proof is much more complicated:

It was proved by constructing a statement that indirectly referred to itself as 'This statement cannot be proved' - to be more precise, it says: 'The $i$-th proposition is not provable'.

By looking just at this sentence, it certainly isn't self-referential, but if we look at how all the propositions were numbered, we can see that $i$ is the number of the above proposition, so the self reference is not direct. Is it what makes the theorem so important and the reason why the proof is so complicated - the fact that statements containing no direct self-reference whatsoever might still refer to themselves indirectly?

Answer 1

Self-reference has a problem, if you want to think about it in terms of "I am not provable" sort of approach. A well-formed formula cannot refer to itself. Moreover, a formula cannot refer to the meta-theory (which is where proofs exist).

What Gödel did was two things:

1. Internalize the meta-theory into the natural numbers via coding, and show that this internalization is very robust.

2. Showed that there is a sentence with Gödel number $n$, whose content is exactly "the sentence coded by $n$ is not provable".

The importance is in both points. They allow us both (limited) access to the meta-theory and the proofs; as well circumvent the problem of being a well-formed formula while still referring to itself. And while the importance of the incompleteness theorem is mainly in the fact that it shows there is no reasonable way to have a finitary proof-verification process to mathematics, and also prove or disprove every sentence; the proof itself is also important because it gives us the internalization of the meta-theory into the natural numbers.

Answer 2

Let the $n$th formula be $F_n(x)$

Godel then takes the formula

$$G(y)\equiv \text{the formula $F_y(y)$ is unprovable}.$$

Now say that $m$ is such that $F_m(x)=G(x)$ then the self referential formula is $$G(m).$$ Basically you are inputing the formulas own Godel number.