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.
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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.
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.
More funny observations could be added.