## Heck’s *Frege’s Theorem* — and *KGFM, 11–14*

When I was in London for the Tennenbaum Workshop, I picked up a copy of Richard Heck’s very recent *Frege’s Theorem*, which collects together eleven of his papers — with some changes and some postscripts — together with a 39 page introductory ‘Overview’. I’ve quickly read the overview which is immensely helpful, as you’d predict, and it is terrific to have the previously very widely scattered papers in one place. Even if you aren’t a great fan of the neo-logicist project, you’ll want to know just how much Frege achieved, and where the pressure points are, technical and conceptual. You won’t get a better guide than Heck. So this collection (the sort of thing that tends to add up to quite a bit more than the sum of its parts) is just great to have, and I really look forward to (re)reading it all.

So there you are — proof positive that I’m not always a cantankerous reader/reviewer! But I’m afraid that I’m again not going to be so friendly about the next four instalments of *Kurt Gödel: Foundations of Mathematics*.

Next up is another piece like Svozil’s that ranges widely over notions of incompleteness in mathematics and science, though at least John Barrow writes very clearly in his ‘Gödel and physics’. He aims at accessibility, but it is all slightly slapdash (from irritating little things like trying to define syntactic consistency using the notion of truth to bigger things like quite mis-stating how a Turing machine is used to decide ‘undecidable’ questions in Mark Hogarth’s now famous construction). So despite the comparative readability, this piece can’t really be recommended to beginners.

The twelfth paper is by Denys Turner, a theologian, on ‘Gödel, Thomas Aquinas, and the unknowability of God’. The author himself thinks that any analogies between Gödel and the tradition of ‘negative theology’ are pretty tenuous, and says “I simply do not know whether the superficial parallel is genuinely illuminating”. Well, it isn’t. Skip this.

The following paper is a really surprising disappointment. I much admire Piergiorgio Odifreddi’s *Classic Recursion Theory* which seems a paradigm of how to write such a book: the exposition is wonderfully clear, but what really makes the book stand out are the historical/conceptual asides about what lies behind the technical developments. I’d have predicted, then, that Odifreddi could have interesting things to say how Gödel’s logical work can be seen as in some way shaped by or encouraged by philosophical ideas. But no: we get less than five pretty superficial pages. Strange.

Finally in this batch, the fourteenth paper — Petr Hájek writing on ‘Gödel’s Ontological Proof and Its Variants’ — may, for all I know, be quite outstanding. Enthusiasts for exploring that strange ‘proof’ will want to read the paper, I’m sure. But I’ve never caught that particular bug: so I frankly confess I’ve just no way of telling how much insightful novelty this is here. Sorry!

OK: that’s taken me over 300 pages through *KGFM*, and so far — Feferman and Rindler apart — I’ve not been enthused. But there’s Hilary Putnam, Harvey Friedman and Hugh Woodin among those yet to come. So I still live in hope!