Berto’s book is cheerfully sub-titled “The complete guide to the incompleteness theorem”. That, I take it, is a joke. But he is perhaps trying to give a sense of how it can be proved in roughly Gödel’s way, which is complete enough to provide a decent platform for ensuing philosophical discussion. And on the whole, I think his basic choice of path here is a good one (see my book, or the cut-down version Gödel Without Tears, which takes a similar route in significantly more detail). But how well does he execute the expository task of filling in sufficient detail?
Well, to continue, Chapter 5 of Berto’s book is about the idea of representing properties, relations and functions in arithmetic, and he states (but only states) the result that all primitive recursive functions are representable in PA. Two quibbles about this short chapter. First, a minor techie point: given Berto’s definition of what it is to represent a function, it isn’t exactly “easy to show” that a property is representalble if and only if its characteristic function is. Second, the box diagram on p. 85 is misleading, for it implies that the representability of general recursive functions in PA is essentially involved in the proof of its incompleteness: not so, of course — it’s the representability of primitive recursive functions that does the job, as Berto says elsewhere.
The next chapter, titled “I am not provable”, does the construction of a Gödel sentence for PA, and gives the syntactic proof of incompleteness. The chapter is disappointing. The basic construction strikes me as being presented in an unnecessarily laboured way: my bet is that the reader who has struggled this far will find this pretty tough. There are smoother presentations on the market! Given this chapter is pivotal, more work could have gone into make this stuff maximally accessible.
Berto has an odd conception of what makes a chapter. I’ve already mentioned his presenting PA and the idea of Gödel numbering in the same chapter. Chapter 7 is another case in point. The first two sections are about how you prove the Second Theorem. Then there’s a crash of gears, and we move to general discussion about the “immediate”, i.e. relatively non-controversial, consequences of the first and second theorems. The obvious presentational glitch here is that Berto officially uses the labels “The First Incompleteness Theorem” and “The Second Incompleteness Theorem” for results specifically about PA. That’s misleading, if only because that’s not how the labels are usually used. But worse, when Berto starts talking a bit about the incompleteness theorems implications (at p. 107), he already has in mind the theorems in the usual sense, i.e. the generalizations to any appropriately strong theories. For instance he writes there “The First Theorem only shows that one cannot exhibit a single sufficiently strong formal system within which all the mathematical problems expressible in the underlying formal language can be decided.” That’s not true of the First Theorem as he has stated it, and it is only five pages later that we meet the generalization that warrants the claim.
Overall, my sense, to be honest, is that the expository half of this book has been put together in a bit of a rush, to get to the interesting philosophical discussions in the second half, and the chapters have not been tried out on enough hyper-critical students and colleagues. Take for example this: “One has (or believes oneself to have) some grasp of the structure N, which one tries to capture by means of the numerals, the syntactic characterizations, etc. [of PA]. However Gödel’s First Theorem entails that this is an illusion.” Grammatically, the “this” ought to refer to the “some grasp” of the structure of N. Well, Gödel’s First Theorem of course doesn’t entail that it is an illusion that you have some grasp. And indeed, it doesn’t entail either that it is an illusion that you in fact have a grasp sufficient to to fix on a determinate structure. The most that can be said is that the idea that you can determinately fully capture the structure of N by a first-order theory like PA is shown to be an illusion: which is probably what Berto meant, but isn’t what his sentences actually say. This sort of casual carelessness is too frequent.
The verdict on the first part of the book, then? Berto’s expository chapters are not too bad, and could (as I said before) initially be helpful for beginners as orientation. But some will struggle as explanations fly past: and those who can cope with the tougher bits should move on to reading a proper presentation. To be continued