Gödel incompleteness revisited

We investigate the frontline of Gödel’s incompleteness theorems’ proofs and the links with computability. The Gödel incompleteness phenomenon Gödel’s incompleteness theorems [Göd31, SFKM+86] are milestones in the subject of mathematical logic. Apart from Gödel’s original syntactical proof, many other proofs have been presented. Kreisel’s proof [Kre68] was the first with a model-theoretical flavor. Most of these proofs are attempts to get rid of any form of self-referential reasoning, even if there remains diagonalization arguments in each of these proofs. The reason for this quest holds in the fact that the diagonalization lemma, when used as a method of constructing an independent statement, is intuitively unclear. Boolos’ proof [Boo89b] was the first attempt in this direction and gave rise to many other attempts. Sometimes, it unfortunately sounds a bit like finding a way to sweep self-reference under the mathematical rug. One of these attempts has been to prove the incompleteness theorems using another paradox than the Richard and the Liar paradoxes. It is interesting to note that, in his famous paper announcing the incompleteness theorem, Gödel remarked that, though his argument is analogous to the Liar paradox, “Any epistemological antinomy could be used for a similar proof of the existence of undecidable propositions”. G. Boolos has proved quite recently (1989) a form of the first incompleteness theorem using Berry’s paradox consisting in the fact that “the least integer not nameable in fewer than seventy characters” has just now been named in sixty-three characters. G. Boolos thought the interest of such proofs is that they provide a different sort of reason for incompleteness. It is true that each of these new arguments gives us a better understanding of the incompleteness phenomenon. When studying proofs and provability, there are two different points of view: the prooftheoretical one (axioms and inference rules) and the model-theoretical one (axioms, models, consequences). The former one tends to be quite syntactical and the latter one more semantical. We have tried to present both points of view and linger over the model-theoretical side because, at least from the author’s point of view, model-theoretic arguments are intuitively clearer than proof-theoretic ones.