In my proof theory monograph there is this exercise:
"The natural proof of PA cannot be carried out in PA. Why? (This proof consists in showing that all theorems of PA are ture.)"
Apparently, by 'natural' proof he means that we accept the (usual) interpretations of the axioms of PA to be true statements and that the rules of the predicate calculus preserve truth (if one likes to consider this a proof).
Isn't the answer simply exactly the second incompleteness theorem? It seemed too easy so I wondered whether there was more to it..
Thanks, Ettore
Here's a link to the question on overflow: https://mathoverflow.net/questions/319417/why-the-natural-consistency-proof-of-pa-cannot-be-carried-out-textbfin-pa