The Wikipedia exposition of Boolos's proof of Godel's first incompleteness theorem assumes that the first incompleteness theorem is equivalent to non-existence of an algorithm that outputs all true sentences of arithmetic and contains no false ones. Also by the end of this proof, it states that "truth outruns proof".
What is the proof of this equivalence?
Godel's incompleteness as usually stated is not about unprovability of a true sentence, it's rather about presence of undecidable statements in the language of the respective theory; that is, there is a statement that the theory in question neither proves nor disproves. And that doesn't seem at first glance to be equivalent to truth outrunning provability.
I mean if theory $T$ is incomplete, then of course we'll have the situation where a true statement is unprovable in it, but the converse doesn't seem to necessarily hold. I mean in principle we may have a theory that doesn't prove every true sentence to be true, yet it can decide on all sentences of it, i.e. it can prove some true sentence (i.e. satisfied by a standard model of arithmetic) to be false. Otherwise the usual Godel proof won't be needing the assumption of $\omega$-consistency to establish the first incompleteness theorem. Since, even without it we do have the usual Godel argument proving the existence of a true sentence, namely the Godel sentence, that the theory doesn't prove. It is known that the $\omega$-consistency assumption is not needed for that part of Godel's proof, it is only needed to establish the unprovability of the negation of the Godel sentence. All in all, it appears to me, that "truth outrunning proof" is weaker than "the theory cannot prove nor disprove some sentence". So, they are not clearly equivalent.
What I mean is that even if one proves that "truth outruns proof", still that doesn't mean that he proved the incompleteness result. So, I don't see how Boolos's argument proves the first incompleteness result. Did he demonstrate the existence of an undecidable statement?
This article answers the above question to the negative. It shows that Boolos's proof is not Rosserable! And that it is equivalent to the same assumptions underlying Godel's original proof, that of a theory being consistent with it's own consistency statement, or simply of a theory not proving it's own inconsistency, a result due to Isaacson.
https://arxiv.org/abs/1612.02549
https://link.springer.com/chapter/10.1007/978-94-007-0214-1_7