Is it possible to reverse engineer an artificial Gödel number encoding that would prove a famous question?

My understanding of Gödel's numbering (?) is that you can use it to uniquely define a mathematical symbol or concept, that encodes the steps to getting there in a number.

https://en.wikipedia.org/wiki/G%C3%B6del_numbering

This allowed Gödel to show a correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers, the key observation of the proof

Can a number be altered, and then it's derivation be worked backwards to find what would need to be different to 'justify' that change?

Say we wanted to prove P=NP, and we can represent P and $G_{n1}$ and NP as $G_{n2}$, and look at the difference in prime factorisation for each.

Could we observe that to set them equal we'd need to make a change to one of the factors?

For instance, if it turned out, actually, $13^9\times G_{n1} = 193^4\times G_{n2}$, that is to set the two concepts as equal we'd need to change their derivation by a factors of $\frac{193^4}{13^9}$ and figure out either how to make that change, or if it meant we could agree on a subset of cases where P=NP.

(n.b. I'm not specifying P=NP as it's vital to the question, it's just the first big topic where this idea would work that came to mind.)

Solution 1:

You're confusing equality of statements and logical equivalences.

The Gödel numbers encode nothing more than the statements, so trying to tweak one statement (P) into the other statement (NP) is usually not the way to prove things.

For example, you wouldn't try to prove Fermat's Last Theorem by looking at the statement and seeing how to tweak it to say "true".