A number of people have now asked me what I think of the fairly new There’s Something About Gödel by Francesco Berto. Well, I’ve been meaning to look at the book anyway: I’m teaching a course on incompleteness in NZ starting in six weeks, and I thought I should know whether I can recommend it as introductory reading for students. So I’ve started the book, and though I plan to crack through at a fast pace, I might as well make a few comments here as I go along.
The book divides into two parts. The first seven chapters are pretty informal exposition (though aiming to give some real sense of how the incompleteness theorems are proved). The last five chapters are more philosophical. I’ll talk about just the first four chapters in this serving of comments.
Chapter One is a whistle-stop tour through some background (introducing e.g. the Liar Paradox and Russell’s Paradox, explaining what an axiomatic theory, what an algorithm is, and so on). This is done with a light touch, but not always with ideal accuracy. The most misleading claim is probably that “the amazing developments of mathematics in the nineteenth century” allowed “the reduction of higher parts of mathematics to elementary arithmetic” (p. 14). Given the informal definitions that precede, it is also misleading to say that every decidable set is enumerable (p. 27). And it won’t really do to say that properties can be considered as sets (p. 23) and then say that Russell’s Paradox shows that not every property “delivers the corresponding set” (p. 32). The concluding pages on set theories (pp. 37-38) are far too fast to be useful — but equally they are not necessary either.
The next chapter starts by outlining Hilbert’s programme. It would unfortunately take a more-than-usually-careful reader to pick up that (1), when doing Hilbertian metamathematics, we are to concentrate on the syntactic features of the relevant formal axiomatized theory we are theorizing about and ignore its semantic properties — rather than it’s being the case that (2) the formal axiomatized theories that Hilbert wants us to study simply lack such properties (are “merely” formal). Berto writes e.g. “In the formalist’s account of these notions, axioms and formal systems are not considered descriptive of anything” (p. 41), which unfortunately sounds like (2). And then the reader will then be puzzled about why, later in the chapter, we are back to talking about semantic features of formal systems.
This chapter also gives first informal presentations of the semantic version of Gödel’s first theorem, and of the second theorem.
Chapter Three both introduces standard first order Peano Arithmetic (I do rather deprecate following Hofstadter and calling it “TNT” for Typographical Number Theory). Most students would I’m sure have appreciated more explanation of the induction schema TNT7 (p. 58) — if the substitution notation has been explained before, then I’ve forgotten that, so I guess that students will too! Also Berto’s half-hearted convention of boldfacing some symbols inside the language of PA but not others is not a happy one: if you are going to do this sort of thing, than use the same face/font choice throughout for all symbols of the formal language. This quite short chapter also introduces Gödel numbering. You’d have to try this out on various student readers, but I’d suspect that this is going a bit too fast for comfort.
Chapter Four is called “Bits of Recursive Arithmetic”; and I think this is rather fumbled. One trouble is that it again will certainly go too fast for most of the intended readers. Berto jumps immediately to a general specification of a definition of one function by primitive recursion in terms of two others, rather than works up to it via a few simple examples: not a good move, in my experience. More seriously, Berto officially sets out to define the full class of recursive functions, mentions minimization, gives no sense of why this keeps us inside the class of computable functions, and then says “I shall skip further development” because “for a proof of Gödel’s Theorem one can take into account only the primitive recursive functions” (p. 75). But then, in the next section, Berto moves on Church’s Thesis which requires us to understand the general notion of a recursive function. Given this isn’t necessary for the incompleteness proof, and that Berto hasn’t actually defined such functions anyway, this would seem to be a recipe for confusion.
Verdict so far: Berto’s exposition in these opening chapters isn’t outstandingly good, but it could initially be helpful for some students as orientation. To be continued