Godel's Proof has been a big and fascinating problem for me for awhile now, but I think I've had a real insight into the problems that it presents. If you'll excuse the improvised formatting, here goes!
1.> <G> <--EQUIVALENT--> <g1 * g2 * g3 ...>
That is, Godel has substituted one term for all the many terms of his argument. All of these smaller terms have Godelian values, which allow them to be made equivalent to a single number.
2.> <NOT> (<g1 * g2 * g3 ...>) <--EQUIVALENT--> <GODEL#>) <--EQUIVALENT--> <UNKNOWN#>
That is, once we break <G> into its separate terms, we see that the <NOT> function must be properly distributed within the statement.
3.a> IF (<--EQUIVALENT-->) has a Godel value, (<--EQUIVALENT-->) must be a term to which the (<NOT>) term can possibly be applied.
3.b> IF (<--EQUIVALENT-->) has no Godel value, then the proof is improperly performed in any case.
4.> (<NOT> (<G> <--EQUIVALENT--> <GODEL#>)) <--EQUIVALENT--> (<G> <--NOT EQUIVALENT--> <GODEL#>)
This is another possible interpretation, based on the arguments above.
In addition to my arguments, I'd like to share some of the thoughts that led me to find, believe and present these conclusions.
Formal logical notation is a form of language, in which we substitute both real material entities and complex concepts with symbols which are taken to be tokens or "pointers" to these entities, and can discuss them at length, and in a formalized procedure, in the case of formal logics. It must be noted, however, that these logics are artificial elaborations on the natural use of language, which roughly began with words, symbols spoken and heard. The physiological act of speaking is a process which has psychological features as well, involving the Broca region of the brain, the vagus nerve, possible changes in heartbeat, respiration, and various other physiological responses, and a dynamic of satisfaction, tension, certainty, or anxiety.
Given the facts above, I hold that intuitive approaches, the beliefs of the logician and the audience, and various other factors are an important part of the "arena" in which the pursuit of logic takes place.