I’ve started writing notes on the expository history, and here’s the first (19 page) instalment. The notes don’t at all aim to be comprehensive, though I’d like to know about significant omissions as I go along. They have been written, as much as anything, as a rather detailed aide-memoire for myself (and a source of bits and pieces I might use). I have done some joining up of the dots to make them tolerably readable, but I certainly haven’t put in the time to spell out everything out in the way a beginning student might want. Still, you shouldn’t need much background to follow the twists and turns.

The notes come in three parts. Part 1 looks at early papers by the Founding Fathers. Part 2, to follow, will look at three pivotal works, Mostowki’s Sentences Undecidable in Formalized Arithmetic and Kleene’s Introduction to Metamathematics (both from 1952), and then Tarski, Mostowski and Robinson’s Undecidable Theories (1953). Part III will continue the story on through some sixty years of textbooks.

Make what use of these notes that you will. Though the usual warning applies in spades, as I’m no historian: caveat lector! Remember how very easy it to LaTeX your work. Just because what you write then looks very pretty doesn’t mean that it is any more authoritative …

In the previous post, I claimed that in his §35 Carnap proves the semantic Diagonalization Equivalence, which he uses in §36 to prove the semantic version of Gödel’s First Theorem. But I said he doesn’t prove the Lemma there or give the now canonical syntactic version of the Theorem (the one depending on the syntactic assumption of -consistency).

Well, no one has protested yet. So, thus emboldened, let me now stick my neck out further!

What happens later in the book? Carnap’s notation and terminology together don’t make for an easy read. But as far as I can see, when he returns to Gödelian matters later, he still is using the semantic Diagonalization Equivalence and not the syntactic Diagonalization Lemma. If the latter was going to appear anywhere, you’d expect to find it in §60 when Carnap returns to the incompleteness of arithmetics: but it isn’t there. (An indication: Carnap here talks of provability being ‘definable’ in arithmetics, and it is indeed expressible — but we know it isn’t capturable/representable by a trivial argument from the Diagonalization Lemma proper. So Carnap hereabouts is still dealing with semantic expressibility, and doesn’t seem to invoke the syntactic notion of capturing/representing needed for the Lemma.)

So, in summary: Yes, Carnap gives a nice tweak to the argument in §1 of Gödel 1931 for the semantic incompleteness theorem, by generalizing the basic idea to give the Diagonalization Equivalence. But this isn’t the modern Diagonalization Lemma. Which isn’t to say that we can’t get the Lemma easily using the wff Carnap constructs: however, as far as I can see, Carnap didn’t explicitly take the step in 1934, even though he is often credited with it.

What am I missing?

Well, in §35 Carnap notes the general recipe for taking a one-place predicate and constructing a sentence such that is true if and only if is true, where as usual is  the formal numeral for the Gödel number for .

Fine. But that claim about constructing a semantic equivalence isn’t the Diagonalization Lemma as normally understood, which is a syntactic thesis, not about truth-value equivalence but about provability. It is the claim that, in the setting of the right kind of theory , then for any one-place predicate we can construct a sentence such that . And Carnap doesn’t prove that in §35.

When we turn to §36, where Carnap proves the incompleteness of his system II, again he appeals to his semantic result not the syntactic Diagonalization Lemma. We now take the relevant predicate to be NOT-PROVABLE: so we get a sentence which is true if and only if it isn’t provable. And then Carnap appeals to the soundness of his system II to argue in the elementary way that isn’t provable. So at this point Carnap is giving a version of the semantic incompleteness argument sketched in the opening section of Gödel 1931 (the one that appeals to a soundness assumption), and not a version of Gödel’s official syntactic incompleteness argument which appeals to -consistency. Indeed, Carnap doesn’t even mention -consistency in the context of his §36 incompleteness proof. He doesn’t need to.

To summarize so far: in §§35–36, Carnap doesn’t use the theorem that his system II proves NOT-PROVABLE, i.e. he doesn’t appeal to an application of the Diagonalization Lemma in the modern sense. He doesn’t need it (yet), and he doesn’t prove it (here).

But maybe he returns to the fray later in the book … the story continues!

Anyway, now that I am working on a second edition, I want to spend the next month pausing to have a look at how others have handled the First Incompleteness Theorem. The basic shape of my book is fixed (after all, I’m doing another edition of IGT, not writing another different book, fun though that would be to do): but I might well get inspiration for how to make local improvements. So how do others state the Theorem? And how do they prove it? I’m planning to write up notes on expository themes and variations as I go along, and will post them here in due course.

The natural place to start is with Hilbert and Bernays. Slight problem: I don’t have German (yep, schoolboy Greek was all well and good, but hasn’t exactly been useful). Does anyone out there have detailed notes of how they prove the First Theorem in §5.1 of their second volume? I’d be really delighted to hear if so! Otherwise, there is a French translation, so I suppose I ought to do battle with that. Not that my French is any good these days either …

Until I can get hold of a translation of Hilbert and Bernays, then, the expository tradition for me will really have to start with Kleene’s wonderful 1952 Introduction to Metamathematics. I’m looking forward a lot to dipping into that again. Then I counted over forty other books on my shelves which give more or less detailed proofs of the First Theorem: hours of fun ahead, obviously. But I’ve started, yesterday and today, by rereading Gödel’s 1931 and his 1934 Princeton lectures. I have to blush again. I’d forgotten, for example, just what Gödel didn’t prove in 1931. ‘We shall only give an outline of the proof’ that every recursive relation is (as we would now say) representable in P;  ’the entire proof’ [that for any predicate expressing a recursive property/relation there is an equivalent arithmetical predicate] ‘can be formalized in the system P‘ [it can be, no doubt, but this isn't proved].

In fact, the situation is this. There are in fact two significantly different results stated in the 1931 paper, the ‘semantic’ and the ‘syntactic’ incompleteness theorems. The first requires that we are dealing with a theory which can express primitive recursive relations and is sound; the second version beefs up the first assumption and requires a theory which can represent p.r. relations while weakening the second assumption to require only that the theory satisfies the syntactic condition of omega-consistency. The first, ‘semantic’ theorem is sketched in Gödel’s §1, the official ‘syntactic’ version is the topic of the central §2 and §3. Look carefully, however, and while there is (enough material for) a full proof of the underplayed semantic version, the proof of the ‘syntactic’ result (the result that is normally meant when people talk now about the First Theorem) is in fact gappy, gappier than I’d really remembered.

Meanwhile, I see over 200 people have downloaded the draft of the first eight chapters of the second edition. Thanks to David Auerbach and Seamus Bradley for some amazingly speedy comments. Keep ‘em coming folks!

OK, here then is

A first instalment of revised chapters of IGT2: Chapters 1 to 8

(these replace the current Chs 1 to 7). I’ve actually rather enjoyed doing this, and am quite pleased with the results — in particular, I think I’ve much improved the too-compressed incompleteness argument in the old Ch. 5. But I’m simultaneously also a bit cheesed off to see how much those early chapters did need improving. It all just goes to show that when you’ve finished writing a book, you should really put it in a drawer for three years until you realize what you were really trying to say, and then re-write it. You can just see your research-driven university approving of that policy, eh?

All suggestions/corrections for these revised chapters will be most gratefully received. You can put them in the comments below, though it is probably better all round to email them (my email address is at the bottom of the “About Peter Smith” page). For copyright reasons, I won’t be able to make all the revised chapters so freely available when I’ve done them: but anyone who emails comments will be put on a circulation list for future tranches of the new version.

Time then to pause to draw breath for Christmas after a busy/distracting time. But then what’s next, logically speaking?

Well, I plan to blog here in the new year about two more books that I’ve been asked to review together — Leon Horsten’s The Tarskian Turn: Deflationism and Axiomatic Truth and Volker Halbach’s Axiomatic Theories of Truth. Not that I claim any special expertise about their topic: but then both books are written for a reader like me — a philosopher/logician interested in theories of truth, who wants to get a handle on recent some formal developments (to which the respective authors have been notable contributors). Should be very interesting. And since both books have been out for few months, I hope some readers of the blog will be able to chip in helpfully in comments!

But mostly, it will have to be back to Gödel. At the moment, I’m re-writing the opening chapters of the book for the second edition, and I think very much improving them — about which I have mixed feelings! On the one hand, it’s very good to feel the effort of doing a second edition is going to be worth while, but on the other hand, I’m a bit downcast to see how very far from ideal those chapters previously were. Sigh. Anyway, when I’ve got the first tranche of chapters more to my liking, I’ll post them here for comments.

And I’ve one or two other plans too … So watch this space!

First up is Angus Macintyre, writing on ‘The impact of Gödel’s Incompleteness Theorems on Mathematics’. His title is pretty much the same as that of a short and very readable piece by Feferman in the Notices of the AMS and his conclusion is also much the same: the impact is small. To be sure, “Some of the techniques that originated in Gödel’s early work (and in the work of his contemporaries) remain central in logic and occasionally in work connecting logic and the rest of mathematics.” But “[a]s far as incompleteness is concerned, its remote presence has little effect on current mathematics.” For example, “The long-known connections between Diophantine equations, or combinatorics, and consistency statements in set theory seem to have little to do with major structural issues in arithmetic” (p. 14). And similarly elsewhere in maths.

There’s a lot of reference to mathematical results, and nearly all of the detailed discussion is well beyond my comfort zone (or that of most readers of this blog, I’d guess: try, e.g., “Étale cohomology of schemes can be used to prove the basic facts of the coefficients of zeta functions of abelian varieties over finite fields”). So I can’t very usefully comment here.

Probably the most exciting and novel thing in this piece is the substantial appendix which aims to give an outline justification for Macintyre’s view that we have “good reasons for believing that the current proof(s) of FLT [Fermat's Last Theorem] can be modified, without abandoning the grand lines of such proofs, to proofs in PA.”  But again, I’m frankly outside my competence here, and I can only refer enthusiasts (or skeptics) about this project to the paper for the details, which look rather impressive to me.

A decidedly tough read for the opening piece!

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!

Or at least, it’s back to Gödel after I’ve written the overdue review of Alan Weir’s book. Watch this space for, belatedly, a few more thoughts on that. Meanwhile, if you’ve bright ideas about how to improve the Gödel book, do let me know!