The paper itself [1] seems quite compact and extremely high level, so I'm sure some heavy hitters would try to reformulate it. Would be the most unintuitive thing to happen since Bell's theorem [2].

[1] https://jhap.du.ac.ir/article_488.html

[2] https://en.wikipedia.org/wiki/Bell%27s_theorem

It wasn't stated why all truths need to be provable though. Perhaps the paper goes into this detail that I'd like explained.

The central claim in particular is not proven because a physical theory P need not be able to express statements like "there exists a number G, which, when interpreted as the text of a theory T, essentially states that the theory T itself is unprovable in the broader physical theory P" as an empirical physical fact.