Gödel’s theorem: larger than proof?

[stefanow]

It's been a decade or so since I've read Nagel & Newman's presentation of Gödel’s famous proof of incompleteness and although not a mathematician or logician myself I couldn’t help but laugh facing an ingenious logical process bound to prove a banality like “I’m lying - it’s true!”. Gödel’s proof is a masterpiece of mathematical and logical demonstration, but I seriously doubt the metaphysical/platonic implications that many people see in it. It’s a formal refutation of formalism, nothing more, nothing less, a mathematic form of Russell’s paradox or liar’s paradox. Gödel didn’t prove that there are certain true but non demonstrable propositions within a certain axiomatic system, but that there is one certain self-indicating proposition like “this proposition is not demonstrable” which is at once non demonstrable but nevertheless true, and he did that by using a powerful version of diagonal method combined with metamathematical (verbal) assumptions in one or two critical points of his demonstration. I haven’t heard of Gödel or anyone else who finally succeeded in demonstrating the same result with non-self-indicating propositions (like, say, Goldbach’s conjecture), which would be extremely interesting and frustrating by the way.

So Gödel showed that there is a certain kind of proposition that is non-demonstrable, but at the same time true within a certain formulated axiomatic system. Obviously (hmm..) such a claim has no meaning with real propositions (propositions that indicate something outside themselves), so Gödel tried it with one self-indicating proposition, which by the way is not a proposition in the common sense and therefore neither true or false as far as our languange game is concerned. It is like saying that something is real but does not exist: the only “something” that corresponds to such a description is nothing. I think that the whole misunderstanding of the significance of Gödel’s proofs derives from his metamathematical use of the word “true” in a context where such a use is nonsensical. Gödel himself used to believe that his proof was something more than a commentary “on formally undecidable propositions of Principia Mathematica and related systems”, so his next logical step was a mathematic proof of the existence of God..

 

[homeomorphic]

Self-reference appears not to be essential and the self referential statements are really only self-referential in concept (that is, outside of the language you are expressing them in).

http://www.math.mcgill.ca/rags/JAC/124/self.html

If only I had time, I would use this as an opportunity to learn this stuff more thoroughly, but unfortunately, I probably don't.

 

[pwsnafu]

I haven’t heard of Gödel or anyone else who finally succeeded in demonstrating the same result with non-self-indicating propositions (like, say, Goldbach’s conjecture), which would be extremely interesting and frustrating by the way.

Goodstein's theorem is the easiest example for a non-mathematician.

 

[Erland]

Gödel showed that the Gödel sentence G is true but not provable within formal arithmetic, provided formal arithmetic is consistent, using that G itself can be interpreted as "G is not provable within formal arithmetic",

But G can be reformulated to say that a certain diofantine equation (a polynomial equation in one or several natural number variables with natural number coefficients ) E has no solutions. If E is provable within formal arithmetic, then the proof itself can be used to construct a solution to E.
So Gödel's Theorem has consequences also for "ordinary" arithmetic, in this case diofantine equations.

Also, we have results such as Goodstein's theorem, which pwsnafu mentioned.

 

[Demystifier]

Godel sentence is self-referential, while the claim of the Goodstein's theorem is not self-referential (or is it?).
My question is: Is there a rigorous way to prove that a sentence (or a set of sentences) is self-referential, or is self-referentiallity a property that can be seen only intuitively?

EDIT:
Perhaps an even better formulation of this question is the following. Can self-referentiality of a sentence in a given axiomatic theory be detected within the theory, without looking at it from a meta-theory?

 

[stefanow]

I think that the self-referential character of G is a metamathematical trait necessary for one in order to be able to apply truth value to G. Otherwise the proposition would be just undecidable (as is proven in the case of Goodstein’s Theorem by Paris-Kirby or the Continuum Hypothesis by Paul Cohen etc.) i.e. neither true nor false within the system’s limitations. Russell had also used self-reference in order to debase Frege’s system and so did Gödel in order to debase Russell’s. I wonder if one could apply some form of Russell's paradox in Gödel’s construction in a manner of “if Gödel’s proof is complete, then excludes itself from its incompleteness claims and thus renders itself incomplete, but this very incompleteness renders it true..”, eventhough Gödel’s proof seems to have been tailored in order to use contradiction for increasing its strength. My memory on the subject could use some refreshment anyway..

 

[Erland]

I think the problem is how we can define "self referential" in a stringent way...

 

[stefanow]

http://www.imm.dtu.dk/~tobo/essay.pdf
This may give an idea of the various forms of self-reference. Note that “Gödel himself had a footnote in his 1931 article… saying that any paradox of self-reference could be used to prove the Incompleteness Theorem”.

 

[stefanow]

Anyway I'd say that the general form of a malicious self-referential sentence could be f(x), only if x=f(x), but would this clear things up?