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In a 7-page article, Chaim Perelman provides an argument supporting the idea that Gödel's premises for the incompleteness theorem bear a contradiction.

Has this ever been refuted? If yes, how? Does anyone have a link to share?

P.S.: I have translated the original article from French to English following a remark in the comments.

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  • Unfortunately, the link you provided is to a document written in French, so judging it would require knowing French, which is going to be an unlikely feature of the average SE contributor. Otherwise, the paper seems to be from 1936, so that we tend to quote Gödel and not Perelman to this day is some testimony that this attempt (like any other) to disprove Gödel has not met with acceptance by the logico-mathematical community at large. Nov 12, 2023 at 23:31
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    FWIW, see Helmer[37] for a repudiation of Perelman vs. Gödel. Nov 12, 2023 at 23:36
  • @KristianBerry Thanks for the Helmer reference. I am wondering whether Helmer is missing that what he calls a 'sentence' is also a mathematical formula in essence. Regarding acceptance, there is a well-known divide between philosopher's usage of logic and the usage of logic in mathematics. Nov 13, 2023 at 1:36
  • AFAIK Helmer isn't missing anything, it's probably Perelman who is. But I was going to try copying Perelman's article into Google Translate to evaluate it myself, but it's not formatted for coherent copying (the drag-and-select function of the cursor won't cover all the text, even one paragraph at a time) so for now I should defer to the weight of intellectual history, not a random outlier. Nov 13, 2023 at 1:56
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    The proof of Gödel’s incompleteness theorems has been verified by many logicians since Gödel’s original paper. In fact, there are proofs that are both simpler and stronger than the original one presented by Gödel. There is little point in trying to pick holes in Gödel’s original formulation. Unfortunately, in the introduction to his paper, Gödel described his proof with rather loose language that encourages some readers to take his remarks out of context and claim errors. See logicmatters.net/igt for a free book on Gödel’s theorems by Peter Smith.
    – Bumble
    Nov 13, 2023 at 6:17

2 Answers 2

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Long comment

You have already Olaf Helmer's review: Perelman versus Gödel (Mind, 1936) and see also S.C.Kleene's review (Journal of Symbolic Logic, 1937) of Helmer.

In addition, you can see a modern exposition of Gödel’s Incompleteness Theorem.

The first key-point is (see Kleene) the "lack of distinction between formulas and Gödel numbers of formulas".

This is evident comparing the modern version of the theorem:

Assuming a consistent formalized system F [ with "suitable" properties] we can construct a sentence G such that: F ⊬ G [in Perelman's paper: ~Dem(qFq) ] and F ⊬ ¬G [ ~Dem(~qFq)].

This sentence, often called “the Gödel sentence” for system F, is an independent, or “undecidable” (that is, neither provable nor refutable in F) formula.

Perelman does not explicitly states consistency, but (page 734) assumes soundness, which is stronger, and this is still correct.

What the author miss is the distinction between the level of the formal system and that of the meta-theory; see Kleene: "distinguish as consistently between the two categories of metamathematical statements and formal mathematical sentences as between those and the third category of syntactical numbers of formal mathematical sentences.

From the two assertions of underivability above, we cannot conclude that the system F proves G ↔ ¬G, that is a contradiction [see formula (10) Dem(~qFq) ≡ Dem (qFq) ].

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The only contradiction I see is, mon ami Kurt the great Gödel proved G (the Gödel sentence) which states of itself (self-reference) that it is unprovable. Is this not a contradiction? If no, wherefore no?

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    G states of itself that it is unprovable in one particular formal system of deduction. We may still prove G in a different formal system, or informally. Godel "proves" it informally. There is no contradiction between being able to prove G informally, and being unable to prove G in one particular formal system.
    – causative
    Nov 13, 2023 at 3:06
  • Copy that. It's just a little weird. Nov 13, 2023 at 3:07

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