The first seven chapters of Berto’s book are exposition of the formalities: I’ve said my piece on those. The last five chapters are philosophical essays, which can be read independently of the particular preceding exposition, as long as you know something about Gödel’s theorems. And indeed the philosophical chapters can be read independently of each other. (As it happens, I read the very last chapter of the book before I read any of the rest of the book, as someone asked me what I thought of the kind of paraconsistent line explored there.)
I’ll take the philosophical chapters in turn, starting with Chapter 8, on “The Postmodern Interpretations”. The first half of this chapter touches on some of the wilder things pomos have said, and some exaggerated claims about Gödelian incompleteness showing that there can’t be a physical theory of everything. Berto briefly says the right kind of things, but he’s largely drawing on Torkel Franzen’s excellent and deservedly familiar demolitions of such daftness, and Franzen does it better.
The second half of the chapter isn’t so much about pomo interpretations in particular as about what we are to make of the phenomenon of non-standard models. There are indeed serious issues here (though of course not all specifically to do with Gödel’s theorems): but Berto’s discussion is too quick and shallow to be useful. And by the end of the chapter he seems to have forgotten the issues that he was supposed to be discussing, namely whether arithmetical statements can sensibly be said to be true only in a relative sense (as in “true-in-the-standard-model). For Berto ends up talking not about deviant models of what we thought was the canonical theory (PA) but about the possibility of deviant theories. So nothing much is achieved here.