I’m getting back down to work on the second edition of An Introduction to Gödel’s Theorems. One thing I plan to do is to put up some pages of exercises as I go along, which I’ve been meaning to do for ages, but takes a surprising amount of time. Watch this space.
Meanwhile, one interim thing I’ve just done is put online a major update to the corrections page for the latest printing(s) of the first edition. The corrections are almost all due to Orlando May, who has evidently been reading the book with a quite preternaturally accurate eye. I’m most grateful.
If anyone else has anything to add, large or small, about how to improve the book next time around, then of course I’ll again be more than grateful!
I haven’t read the book, so I can’t say anything about it for sure. But judging from what’s online, it looks like the book goes through the usual arguments to get to the theorems. Is that correct?
I’m interested more in the implications of the theorems. So far as Goedel’s first theorem is concerned, I don’t see what the big deal is. It’s only shocking if one assumed that arithmetical truths could be axiomatized to begin with. But if you never assumed that, it doesn’t seem like such a big deal.
On the other hand, Voevodsky of the IAS takes Goedel’s second theorem to indicate that arithmetic is actually inconsistent. That would be a big deal.
The book does indeed go though the usual arguments (it is, after all, an Introduction).
Is the first theorem a “big deal”? Not really, as far as working mathematics (outside logic) is concerned: see e.g. Feferman’s readable short paper The impact of the incompleteness theorems on mathematics.
But it’s a fairly big deal philosophically. It tells us that we can’t equate arithmetic truth with provability in some fixed nicely axiomatized system. But then what does make for arithmetical truth?
As to Voevodsky: the discussions I’ve seen of this on FOM suggest he is barking up an unpromising tree.