Gödel’s theorems


GWT2 — a third (near-final?) full draft

There is now a third complete draft of the forthcoming new edition of Gödel Without (Too Many) Tears. You can download it here.

What’s changed this time, since the last full draft? There has been some more typographical micro-adjusting (you won’t notice!). A few more typos have been fixed, and there have been some scattered very minor changes in phrasing for clarity’s sake. I plan to do a bit more work on the index, but I hope the rest of the book is now in a near-final state.

Corrections and suggestions for local improvements will still be extremely welcome for another few weeks. I’ll then be getting back to GWT after a planned holiday and family time. To be definite: I’ll be calling a halt to further tinkering the weekend of October 29th, with the aim of getting a second edition out in print in November. So all comments, including — especially including! — quick notes of the most trivial typos, will be most welcome until then. (And many thanks to those who have emailed comments so far.)

Added Sept. 25: Minor corrections/revisions in Ch. 14, Ch. 17 and Appendix.

GWT2 — a second full draft

There is now a second third complete draft of the forthcoming new edition of Gödel Without (Too Many) Tears. You can download it here.

What’s changed? There has been a fair bit of typographical tidying (which you no doubt won’t notice, but I might as well try to make things consistent!). The index has grown a bit, though there is more work to be done there. Some typos have been removed, there have been some scattered minor changes in phrasing, and further changes to tidy the way topics in different chapters are linked together. But the main update has been to the chapter on the Diagonalization Lemma: I’ve hopefully much improved this by re-arranging the material in a more logical way.

So that’s enough by way of updating to be worth putting on line now. The first full draft has been downloaded about 650 times. If you are one of those actually reading it, you might want to download the new improved version.

To repeat what I said before: It is too late to write a very different book, and after all this is supposed to be just a revised edition of the seemingly quite well-liked GWT1! This is not the moment, then, for radical revisions. But otherwise, all suggestions, comments and corrections, including quick notes of the most trivial typos, will be most welcome! Send to the e-mail address on the first page of PDF, or comment here. (Just note the date of the version you are commenting on.) Comments will continue to be welcome for the next month or so.

Added Sept 17: revised version of second draft posted, with minor changes to Preface and Chapters 1 to 4.
Added Sept 18: revised version posted, this time with minor improvements to Chapter 5.
Added Sept 21: revised version posted, with mostly very minor improvements to Chapters 6 to 15 (the one substantive correction is at the very top of p.76 where I’d made an unnecessary assumption that we are dealing with a consistent theory).
Added Sept 22: second draft now replaced by a third draft

The Annotated Gödel

Some years ago, Charles Petzold published his The Annotated Turing which, as its subtitle tells us, provides a guided tour through Alan Turing’s epoch-making 1936 paper. I was prompted at the time to wonder about putting together a similar book, with an English version of Gödel’s 1931 paper interspersed with explanatory comments and asides. But I thought I foresaw too many problems. For a start, not having any German, I’d have had to use one of the existing translations, which would lead to copyright issues, and presumably extra problems if I wanted to depart from the adopted translation by e.g. rendering Bew by Prov, etc. And then I felt it wouldn’t be at all easy to find a happy level at which to pitch the commentary.

Plan (A) would be to follow Petzold, who is very expansive and wide ranging about Turing’s life and times, and is aiming for a fairly wide readership too (his book is over 370 pages long). I wasn’t much tempted to try to emulate that.

Plan (B) would be write for a much narrower audience, readers who are already familiar with some standard modern textbook treatment of Gödelian incompleteness and who want to find out how, by comparison, the original 1931 paper did things. You then wouldn’t need to spend time explaining e.g. the very ideas of primitive recursive functions or Gödel numberings, but could rapidly get down to note the quirks in the original paper, giving a helping hand to the logically ept so that they can navigate through. However, the Introductory Note to the paper in the Collected Works pretty much does that job. OK, you could say a bit more (25 pages, perhaps rather than 14). But actually the original paper is more than clear enough for that to be hardly necessary, if you have already tackled a good modern treatment and then read that Introductory Note for guidance in reading Gödel himself.

Plan (C) would take a middle course. Not ranging very widely, sticking close to Gödel’s text. But also not assuming much logical background or any prior acquaintance with the incompleteness theorems, so having to slow down to explain ideas of formal systems, primitive recursion and so on and so forth. But to be frank, I didn’t and don’t think Gödel’s original paper is the best peg on which to hang a first introduction to the incompleteness theorems. Better to write a book like GWT! So eventually I did just that, and dropped any thought of doing for Gödel something like Petzold’s job on Turing.

But now, someone has bravely taken on that project. Hal Prince, a retired software engineer, has written The Annotated Gödel, a sensibly-sized book of some a hundred and eighty pages, self-published on Amazon. Prince has retranslated the incompleteness paper in a somewhat more relaxed style than the version in the Collected Works, interleaving commentary intended for those with relatively little prior exposure to logic. So he has adopted plan (C). And the thing to say immediately — before your heart sinks, thinking of the dire quality of some amateur writings on Gödel — is that the book does look entirely respectable!

Actually, I shouldn’t have put it quite like that, because I do have my reservations about the typographical look of the book. Portions of different lengths of a translation from the 1931 paper are set in pale grey panels, separated by episodes of commentary. And Prince has taken the decidedly odd decision not to allow the grey textboxes containing the translation to split themselves over pages. This means that an episode of commentary can often finish halfway down the page, leaving blank inches before the translation continues in a box at the top of the next page. And there are other typographical choices while also unfortunately make for a somewhat  unprofessional look. That’s a real pity, and does give a quite misleading impression of the quality of the book.

Now, I haven’t read the book with a beady eye from cover to cover; but the translation of the prose seems quite acceptable to me. Sometimes Prince seems to stick a bit closer to the Gödel’s original German than the version in the Works, sometimes it is the other way about. For example, in the first paragraph of Gödel’s §2, we have

G, undefinierten Grundbegriff: W, primitive notion: P, undefined basic notion,

while in the very next sentence we have

G, Die Grundzeichen: W, The primitive signs: P, The symbols.

But such differences are relatively minor.

Where P’s translation of G departs most is not in rendering the German prose but in handling symbolism. W just repeats on the Englished pages exactly the symbolism that is in the reprint of G on the opposite page. But where G and W both e.g. have “Bew(x)”, P has “isProv(x)”. There’s a double change then. First, P has rendered the original “Bew”, which abbreviated “beweisbare Formel”, to match his translation for the latter, i.e. “provable formula”. Perhaps a good move. But Prince has also included an “is” (to indicate that what we have here is an expression attributing a property, not a function expression). To my mind, this makes for a bit of unnecessary clutter here and elsewhere: you don’t need to be explicitly reminded on every use that e.g. “Prov” expresses a property, not a function.

Elsewhere the renditions of symbolism depart further. For example, G has “A_1\mbox{-}Ax(x)” for the wff which says that x is an instance of Axiom Schema II.1. P has “isAxIIPt1(x)”. And there’s a lot more of this sort of thing which makes for some very unwieldy symbolic expressions that I don’t find particularly readable.

There are other debatable symbolic choices too. P has “\Rightarrow” for the object language conditional, which is an unfortunate and unnecessary change. And P writes “x[v \leftarrow y]” for the result of substituting y for v where v is free in x. This may be a compsci notation, but to my untutored eyes makes for mess (and I’d say bad policy too to have arrows in different directions meaning such different things).

Other choices for rendering symbolism involve more significant departures from G’s original but are also arguably happier (let’s not pause to wonder what counts as faithful enough translation!). For example, there is a moment when G has the mysterious “p = 17 Gen q”: P writes instead “p = forall(\mathbf{x_1}, q)”. In G, 17 is the Gödel number for the variable x_1: P uses a convention of bolding a variable to give its Gödel number, which is tolerably neat.

There’s more to be said, but I think your overall verdict on the translation element of Prince’s book might go either way. The prose is as far as I can judge handled well. The symbolism is tinkered with in a way which makes it potentially clearer on the small scale, but makes for some off-putting longwindedness when rendering long formulas. If you are going to depart from Gödel’s symbolism, I don’t think that P chooses the best options. But as they say, you pays your money and makes your choice.

But what about the bulk of the book, the commentary and explanations interspersed with the translation of Gödel’s original? My first impression is definitely positive (as I said, I haven’t yet done a close reading of the whole). We do get a lot of helpful framing of the kind e.g. “Gödel is next going to define … It is easier to understand these definitions if we think about what he needs and where he is going.” And Prince’s discussions as we go along do strike me as consistently sensible and accurate enough, and will indeed be helpful to those who bring the right amount to the party.

I put it like that because, although I think the book is intended for those with little background in logic, I really do wonder whether e.g. the twenty pages on the proof of Gödel’s key Theorem VI will gel with those who haven’t previously encountered an exposition of the main ideas in one of the standard textbooks. This is the very difficulty I foresaw in pursuing plan (C). Most readers without much background will be better off reading a modern textbook.

But, on the other hand, for those who have already read GWT (to pick an example at random!), i.e. those who already know something of Gödelian incompleteness, they should find this a useful companion if they want to delve into the original 1931 paper. Though some of the exposition will now probably be unnecessarily laboured for them, while they would have welcomed some more  “compare and contrast” explanations bringing out more explicitly how Gödel’s original relates to standard modern presentations.

In short, then: if someone with a bit of background does want to study Gödel’s original paper, whereas previously I’d just say ‘read the paper together with its Introductory Note in the Collected Works’, I’d now add ‘and, while still doing that, and depending quite where you are coming from and where you stumble, you might very well find some or even all of the commentary in Hal Prince’s The Annotated Gödel a pretty helpful accompaniment’.

GWT2 — a first full draft

I now have put together a first complete [updated, now third] draft of the second edition of Gödel Without (Too Many) Tears. You can download it here.

I need to do a careful read-through for typos/thinkos. I also need to update the index, make the typography more uniform between chapters, and e.g. decide on a more consistent policy about when I cross-reference to IGT2. That sort of fun to come over the next month or so. I’ll post updates from time to time, and the link above will keep pointing to the current draft version of GWT2.

It is too late to write a very different book, and after all this is supposed to be just a revised edition of the seemingly quite well-liked GWT1! This is not the moment, then, for radical revisions. But otherwise, all suggestions, comments and corrections, including quick notes of the most trivial typos, will be most welcome! Send to the e-mail address on the first page of PDF, or comment here. (Just note the date of the version you are commenting on.)

Actually, many readers of this blog will have better things to do than spend much time with this sort of intro-level enterprise (though massive thanks are due to a handful who have already been e-mailing comments). But if you aren’t a student yourself, you could well have students who would be interested to take a look and let me know what they find too obscure, and/or give other comments and corrections. (Back in the day, I lined up a whole team of volunteers to look at a couple of chapters each of IGT2: and there turned out to be precisely zero correlation between “status”, from undergraduate to full professor, and the usefulness of comments!) So do please spread the word to any students, undergraduate or graduate, who might be — or ought to be! — interested.

It’s not much of a bribe, I know, but those impoverished students who prompt the biggest corrections/improvements will get a free paperback in due course, as well as having their name in lights in the Preface!

Added: Latest version, linked above, Thursday 12 September.

GWT2 — up to the first incompleteness theorem

I have now revised Gödel Without (Too Many) Tears up to and including the pair of chapters on the first incompleteness theorem. You can download the current version up to Chapter 13 here.

For info: the chapter on quantifier complexity has been revised (adopting a more complex definition of Sigma_1 sentences, so that I don’t have to cheat later in saying that primitive recursive functions can be defined by Sigma_1 sentences). Then the chapter on primitive recursive functions has been slightly revised yet again. I have tried to make the chapter that proves that primitive recursive functions can indeed be defined by Sigma_1 sentences a bit more reader-friendly (the key ideas are elegantly simple: implementing them is unavoidably a bit messy). The chapter on the arithmetization of syntax is little altered. And finally in this instalment, the two chapters on the semantic and syntactic versions of the first incompleteness theorem are more or less untouched.

I’m still on track for getting a second edition out by around the end of October. It goes without saying that all comments and corrections will be gratefully received (and do please alert any students who might be interested in reading through and spotting typos or unclarities). Many thanks once again to David Furcy and Rowsety Moid for corrections and suggestions.

Added 24 Aug Corrected version uploaded.

IGT as a free download, two years on …

I’ve just noticed that it is almost exactly two years since I was able to make An Introduction to Gödel’s Theorems, originally published by CUP, freely available to download as a PDF.

After a ridiculously large initial flurry of downloads, the book is now steadily downloaded about 600 times a month. As I’ve said before about such stats, it is very difficult to know how to interpret the absolute numbers: but this looks respectable enough, and  the trend is still upwards.

When I originally announced, without much fanfare, that the PDF was available, I added

I may in due course also make this corrected version of the book available as an inexpensive print-on-demand book via Amazon, for those who want a physical copy. But I doubt that there would be a big demand for that, so one step at a time

Well, of course, I soon did set up the POD paperback (very easy if you already have a publication-quality PDF), and I was proved quite wrong about demand. I thought any sales would be a tiny trickle given the availability of a completely free download; but in fact the paperback of IGT very steadily sells over 50 copies a month, over three times as many it was doing under the auspices of CUP. So I think we can count the experiment as a success!

Back in the day, when I was writing the first edition of IGT, I tried to do the whole thing from memory, reconstructing proofs as I went, on the principle that if an idea or a proof-strategy had stuck in my mind, then it was probably worth including,  and if it hadn’t then maybe not. I did fill in some gaps once I had a complete good draft; but that explains the relative shortage of footnotes to sources. When I look at the book occasionally, it is just a tad depressing to realize that I would struggle to rewrite it from memory now. That struck me forcefully yesterday when, reworking a section in Gödel Without (Too Many) Tears for its second edition, I consulted IGT and came across an important point that, at least for the moment, I’d quite forgotten the intricacies of. Ah well …

GWT2 — another instalment

I have now revised three more segments of Gödel Without (Too Many) Tears. I’m still aiming to get a second edition out by the end of October.

So here is the whole first third of what will be GWT2, now taking us up to and including the first Interlude. The chapter introducing Peano Arithmetic has been quite significantly revised from the current printed version; the chapter on Quantifier Complexity gets some minor tweaks; and the following Interlude is hardly changed at all. Many thanks once again to David Furcy for corrections and suggestions.

I‘m not entirely sure, as I read through, that I’d write the book quite the same way if I were starting all over again from scratch. But no matter: the plan is to improve the current book, not to write a different one!

GWT2 — a third instalment

I was prompted to start working on a second edition of Gödel Without (Too Many) Tears by discovering that there was a significant muddle in Chapter 5 (I had inconsistently wavered between taking Baby Arithmetic, so called, as having just a negation connective, and having other connectives too). Thanks again to Ben Selfridge for pointing out that embarrassing glitch.

This new instalment of GWT2 corrects that unfortunate mess, and makes a number of other small improvements for clarity/readability, in what is now numbered as Chapter 6. So here it is!

To save readers having to dart between different PDFs, I have included the revised versions of the Preface and earlier chapters (with only trivial changes from the previous posting).

Need I say it? No: but I still will! Corrections and friendly suggestions are immensely welcome at this stage.

Already updated with many thanks to David Furcy and Léon Probst. (You may need to force your browser to reload the file to get the version dated 18 July.)

GWT2 — a first instalment

I’ve decided to get to work, putting together a revised version of Gödel Without Too Many Tears. 

I’ll obviously correct the known glitches in the first edition. But as I start to read through the opening chapters, I find myself wanting to make quite a lot of little stylistic improvements. And there will be a couple of more significant changes, I think, later in book. So this will be, while not a radically different version, still rather more than a lightly corrected reprint. So we’ll count  it as a second edition, GWT2.

I’ll post revised chapters here in bite-sized instalments, from time to time over the next few weeks. As always, corrections and friendly suggestions are immensely welcome at this stage: now is the time to let me know if you had issues with the first edition.

Here then are the first fourteen pages, based on the old Chapters 1 and 2, now three chapters. Enjoy!

Added 9 July: With many thanks to David Furcy, I’ve already uploaded a corrected version, repairing about ten(!) minor typos.

Juliette Kennedy, Gödel’s Incompleteness Theorems

I was in the CUP Bookshop the other day, and saw physical copies of the Elements series for the first time. I have to say that the books are suprisingly poorly produced, and very expensive for what they are. I suspect that the Elements are primarily designed for online reading; and I certainly won’t be buying physical copies.

I’ve now read Juliette Kennedy’s contribution on Gödel’s Incompleteness Theorems. Who knows who the reader is supposed to be? It is apparently someone who needs the notion of a primitive recursive function explained on p. 11, while on p. 24 we get a hard-core forcing argument to prove that “There is no Borel function F(s) from infinite sequences of reals to reals such that if ran(s) = ran(s’), then F(s) = F(s’), and moreover F(s) is always outside ran(s)” (‘ran’ isn’t explained). This is just bizarre. What were the editors of this particular series thinking?

Whatever the author’s strengths, they don’t include the knack of attractive exposition. So I can’t recommend this for reading as a book. But if you already know your way around the Gödelian themes, you could perhaps treat this Element as an occasionally useful scrapbook to dip into, to follow up various references (indeed, some new to me). And I’ll leave it at that.

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