Is Gödel's theorem invalid? [closed]

Question

Right now I've skim through Gödel's theorem is invalid by Diego Saá on arXiv (freely available).

As it seems very plausible, I ask for any references and scrutinizations of the paper.

Answer 1

The first incorrect claim I noticed was in formula (12), where it is claimed that

$$(\forall x)Bew(P(x)) \to Bew((\forall x)P(x))$$

That is not correct. A counterexample is obtained by considering the standard model of arithmetic and letting $P(x)$ say "$x$ is not a coded proof of $0=1$". For every standard natural number $n$, PA does prove this statement $P(n)$; but $(\forall x)P(x)$ is exactly the statement Con(PA) that is not provable in PA by the incompleteness theorem, so the right side is false in the standard model. So the implication in (12) fails in the standard model.

This sort of mistake is easy to make, and it is one reason that it is important to use Gödel number markers; the statement should be written $$ (\forall x)Bew(\ulcorner P(x) \urcorner) \to Bew(\ulcorner (\forall x)P(x)\urcorner), $$ The Gödel number markers make it clear which things are actually formulas in the metatheory and which are just numbers that the object theory uses to encode formulas. They also help the reader remember that independent proofs of $P(n)$ for every $n$ do not directly combine to make a proof of $(\forall x)P(x)$, as (12) claims they do.

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P.S. Another tell-tale sign that a paper on the incompleteness theorems is fishy is when the author refers solely to Gödel's original paper, as if there had been no other published proof of the incompleteness theorems in the meantime. The original paper by Gödel was groundbreaking, but it is written in archaic terminology that would not be in any modern book. Any sound critique of the incompleteness theorems would need to engage with the modern expositions of the proof, not just the original one.

Answer 2

1. No mathematician would explicitly make a copyright statement right under the title of such an astounding essay.

2. The heading structure of the document is horrible.

3. The proof in the appendix that the induction principle need not be postulated as an axiom but rather is a theorem replaces logic with "growing sets of elements" and mere "and so on" handwaving. The axiom (scheme) of induction is precisely a replacement of such doubtful methods. If one wants to make section 10 rigorous, one needs induction :)

4. There goes that "potential vs completed infinity" discussion again. Growing sets can hardly be called a well-defined concept. But if at all, this paper shows that in the author's different foundation of mathematics the Gödel theorem does not hold.

5. This, let us say "different", foundation has a very, let's say "special", interpretation of the allquantor $\forall$.

6. The claim in section 2 that the induction principle is equivalent to $\forall x P(x)\to\forall x P(x)$ is obviously false. The latter tautology holds also in models that do not allow induction.

7. On page 9 he writes "Take notice that the relation is not demonstrated and it is very strange to mathematics.". Actually, the relation is demonstrated and one can write down very explicit self-referential statements.

8. The claim that (32) was not proved by Gödel is not feasible, especially as it comprises a basic rule of inference.

9. The complaint that Gödel uses $(33)$ only implicitly without prior mention is ridiculous, given the fact that $\neg (33)$ allows us to prove anything and the alternative that neither $(33)$ nor $\neg(33)$ would make $(33)$ itself undecidable.

10. The "inconsistency" on page 14 appears immediately after introducing the unsound reasoning from (12).

11. Next the author talks about wishing to "suppress" some statements in "our formal systems". Since the statements were obtained by valid rules of inference, this is a daunting task.

12. The formulation about the beginning of section 6 is all wrong. There are axiomatizatons of PA that are complete and consistent (or at least as consistent as the incomplete axiomatizations are), e.g. let every theorem be an axiom. The crucial thing is that the axiomatization should be recursively enumerable. The "accidental" completion of axiomatization of PA in a computer is thus nonsense.

More funny observations could be added.