Gödel's Proof as a proof strategy for P = NP or P != NP I have had this thought for quite a while. Gödel proved the incompleteness of arithmetic by creating a one-to-one correspondence with a number and certain numerical relationships to create a statement that says, in effect, "I cannot be proven." 

My question is: has there been any attempt to go the other direction? That is, has anyone succeeded in showing that an arbitrary statement -- such as P=NP -- Is a statement that says of itself that it has no proof? In order for this to happen, a one-to-one relationship needs to be shown between what that means in the mathematics of it and its "meta meaning". In short, Gödel went from meta-meaning ("this statement cannot be proven") and constructed a number that, in effect, stated this meta-meaning. Has there been attempts to show that statements already in existence have a meta meaning stating that they cannot be proven? if so, has anybody used this strategy to prove or disprove P=NP's provability? It may be a simple question, but it has been bothering me for a while. Thanks for reading :)
My thoughts on the subject, which must be taken with a grain of salt as I'm far from being an expert on this subject (or any subject, to that matter):

This idea was mentioned in Douglas Hofstadter's "Godel, Escher, Bach", but I never saw it seriously discussed. As far as I know, no independence proof (e.g. the independence of the Continuum hypothesis) has ever used such a technique, and it's hard to see why P=NP will be different.

It is worth remembering that Godel's statement is difficult to build. In effect, you can think of it as a logical encoding of a small computer program which knows how to decode Godel's numbering and knows how to check proofs (in an effective system, i.e. one in which proof checking is computable), and to top it all, it also contains an encoding of itself using a smart diagonalization trick. This is not something you stumble upon; it is a very delicate construct. 

I see no reason to assume that a statement such as P=NP, which deals with a class of Turing machines, can be re-interpreted as discussing itself. If such a far-fetched interpretation can be made for P=NP, why can't it also be done for, say, IP=PSPACE (which is well known to be true)?

So although this idea is quite fascinating, I see no reason to believe it is true; however, it might be explored by taking some toy independent statements and trying to see if they can be re-interpreted in a way they talk about themselves (I think it will already prove immensely difficult because of the complexity of making a statement talk about itself). I would start there, not attack P=NP directly.

It is also worth mentioning that there is no apparent reason why P=NP is undecidable. There are many indications that it is undecidable using standard methods (e.g. diagonalization - see Baker-Gill-Solovay) but many other complexity-theory problems had a similar difficulties and were laid to rest using a clever new formulation or point of view (IP=PSPACE being a nice example).
I am far from an expert also, but every independent statement can be interpreted as saying 'I am not provable', thats what it means to be undecidable. And every theorem says 'I am provable'.