So, at long last, I’ve actually sent off to CUP the PDFs for the reprint of An Introduction to Formal Logic. And no, I’ve not been spending months non-stop on the revisions. But overall, I have now spent quite a bit of time on this; and I think it’s been well worth the effort. It’s still the same book, and I guess that there aren’t really enough changes to make this count as a forthcoming new edition. But there are some new paragraphs here and there, and there’s a large number of small changes in phraseology — not to mention, of course, the correction of known typos/thinkos in the first printing. The cumulative effect is a book I’m markedly happier with — even if it’s not the book I’d be writing if were starting over from scratch. Still, I’m glad that’s done.

Another matter I’ve been turning over in my mind more seriously in the last month or so is the question of which book to write next. There’s a number of possibilities, including an intermediate logic text following on from the Introduction. I did at one point say I was going write a sequel to the Gödel book — as it might be, Incompleteness After Gödel. But the more I’ve been thinking about that, the more dauntingly wide the field seems to become. So I’m now minded to be much less ambitious, and to write around and about just one theme, namely proofs of the consistency of arithmetic. In particular, what do we make of the significance of Gentzen’s various proofs? Indeed, how do those proofs actually work? (Not that I want to write a primarily historical book — I’ve not got the skills or the knowledge for that. I’m after neat rational reconstructions.) I’ve a ton of work to do to get more on top of this stuff. But it should be fun.

Meanwhile, for light relief, I must get back to Charles Parsons’s book, as I’ve a review to write in the next few weeks. So over the coming days it will be back to blogging about that.

(Oh, I’m sure that other congenital procrastinators will recognize the self-motivating technique here! If you tell the world you are going to do X — where X is something you really quite want to do anyway, but fear you might be distracted from — then avoiding the sheer embarrassment consequent on publicly not doing X now becomes a big added reason to stop faffing around and get down to it. So … to work!)

Ole Thomassen HjortlandA book on Gentzen’s proofs sounds like a great idea. In fact, we (i.e., me and some other logic-minded people in Melbourne) have thought about having a reading group on Gentzen’s proofs. But we’re struggling a bit with where to start. Do you have any good suggestions for reading material?

Peter SmithHi Ole! I don’t really have good suggestions — indeed it was the lack of obvious things to recommend to a grad student reading group here, together with my (failed!) struggles to produce some coherent notes, that in part got me thinking about taking a more serious look at this stuff and trying to write a book.

a cartesianA book on Gentzen could be very interesting, but I have to say I’d have looked forward more to Incompleteness After Gödel.

I don’t know of anything that would give someone who knew the basics of incompleteness from your book or from a logic course a good feel for how incompleteness is viewed now or for what impact it’s had, if any, on mathematics generally.

There is a lot of technical work in set theory, some covered in books such as Kanamori’s The Higher Infinite, but I don’t know of anything at a much more accessible level.

Eric ThomsonJust a vote that it would be great to see work on Gentzen for those with knowledge on a level in your first Godel book. I recently ran into Gentzen’s proof and realized how little I know about the topic of consistency.