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.(adsbygoogle = window.adsbygoogle || []).push({});

So Gödel showed that there is a certain kind of proposition that is non-demonstrable, but at the same timetruewithin 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..

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# Gödel’s theorem: larger than proof?

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