Craig Smorynski reviews IGT

I hadn’t noticed that Craig Smorynski has reviewed my Introduction to Gödel’s Theorems in Philosophia Mathematica — thanks to Eckart Menzler for alerting me.

Smorynski says some very nice things:

The goal of the book is not to introduce the reader to Gödel’s Theorems, which can be done in far fewer pages than the roughly 350 pages of the book, but to introduce the whole problem area initiated by the appearance of Gödel’s paper. The author succeeds in this broader goal as well as can be expected.


Smith does as thorough a job in presenting and proving Gödel’s Theorems as one could hope for in a book of this type. I do not hesitate to recommend the book as an introduction to the whole circle of Gödel’s Theorems and the issues surrounding them to the major portion of Smith’s audience: the ‘philosophy students taking an advanced course’. For them, Smith’s book is about as good as it gets.

Given how much I admire Smorynski’s own classic presentation in his Barwise Handbook article (and of course other work of his), this is praise indeed, and I’m delighted! Let me just add a few comments on some of his comments.

  1. I did advertise the book as not just being suitable for philosophers but also for ‘mathematicians who want a more accessible exposition’. I was being hopeful, of course! Smorynski says “I cannot whole-heartedly recommend the book” for them. The book is too leisurely, and there too many philosophical asides. Well, I didn’t say that all mathematicians will like it! — but judging from the occasional email, some do like the pace and the asides. But yes, mathematicians will just have to look to see if it works for them — there are plenty of  alternative presentations with a more “hard core mathmo” style.
  2. In Chap. 6, I give an incompleteness theorem (with a weaker conclusion than Gödel’s) that I said was “due to” Tim Smiley — I learnt it from him in lectures, and he told me that he’d worked it out himself. I also said that, as far as I knew, the first published version was by Hunter in his Metalogic 1971. Smorynski reports that the same argument can in fact be found in Michael Arbib’s 1964 book Brains, Machines, and Mathematics.
  3. “Speaking as a mathematician, I gleefully note that Smith’s attempt on page 16 to snipe at mathematicians … backfires.” That puzzled me. There is no sniping. I did say that it’s “a trivial arithmetical puzzle” to write down a pairing function that zig-zags through the ordered pairs of numbers. I meant: it involves nothing conceptually difficult. Which is true, and snipes at no one! :-)
  4. “Smith’s statement of the Chinese Remainder Theorem in footnote 6, p. 112, is not a statement of the Chinese Remainder Theorem, but of the key idea behind a popular combinatorial proof of the same, a proof that in fact does not formalise directly in PA without one’s already having shown how to deal in some other way with finite sequences.” Oops. Yes, the labelling is indeed careless. And the point that the proof doesn’t formalize unless we are already able to deal with finite sequences is interesting when it comes to the second theorem.
  5. In a footnote on p. 78, I say “To be strictly accurate, Presburger—then a young graduate student— proved the completeness not of P but of a slightly different and less elegant theory (whose primitive expressions were ‘0’, ‘1’ and ‘+’).” I confess I was relying on second-hand information. Smorynski puts me right: “Presburger’s paper proved the completeness and decid ability of the theory of addition of the integers, including the negative ones. This is a genuinely easier result. However, in the same volume in which Presburger did this, he also published an addendum announcing, without proof, that the results carried over when the order relation was added as a primitive. Hilbert and Bernays, 1934, presents the first published proof [for P] along Presburger’s lines.”
  6. Finally, on p. 210 (p. 211 in the current printing), I write “The first example of a more natural, non-Gödelian, arithmetical statement which is true, statable in the language of basic arithmetic, yet demonstrably not provable in PA, was found by Jeff Paris and Leo Harrington in 1977.” I’m more than a bit embarrassed that this survives into the fourth printing, for Smorynski isn’t the first to point out to me that, as in a sense I knew perfectly well, “This is simply false. … The first ‘natural, non-Gödelian arithmetical statement’ unprovable in PA was induction up to ε0, shown underivable by Gentzen in his Habilitationschrift in 1939. Following that, there was the unprovability of the totality of certain recursive functions as emphasized by Kreisel circa 1958.” Indeed.

Still, Smorynski’s overall verdict remains pleasingly positive.

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