Gödel’s theorems

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Logicisms and Gödel’s Theorem

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Russell famously announced “All mathematics deals exclusively with concepts definable in terms of a very small number of logical concepts and … all its propositions are decidable from a very small number of fundamental logical principles.” That wildly ambitious version of logicism is evidently sabotaged by Gödel’s Theorem which shows that, fix on a small number of logical principles and definitions as you will, you won’t even be able to derive from them all arithmetical truths let alone the rest of mathematics. But how do things stand with latter-day, perhaps less ambitious, forms of neo-logicism?

Second-order logic plus Hume’s Principle gives us second-order Peano arithmetic, and True Arithmetic is a semantic consequence. Buy, for the sake of argument, that Hume’s Principle is in some sense as-good-as-analytic. But how does that help with the epistemological ambitions of a logicism once we see that Gödel’s Theorem shows that second-order semantic consequence is not axiomatizable? Fabian Pregel at Oxford has a very nice piece ‘Neo-Logicism and Gödelian Incompleteness’ coming out shortly in Mind, arguing first that the earlier Wright/Hale canonical writings on their neo-logicism unhappily vacillate, and then that Wright’s more extended discussion of the issue in his 2020 ‘Replies’ (in the volume of essays on his work edited by Alexander Miller) is also unsatisfactory.

I’ll leave Pregel to speak for himself, and just recommend you read his piece when you can. But I was prompted  to look again at Neil Tennant’s recent discussion of his deviant form of neo-logicism (which isn’t Pregel’s concern). Right at the outset of his The Logic of Number (OUP, 2022), Tennant is emphatic that (a) respecting what he calls the Gödel phenomena must be absolutely central in any sort of foundationalist story. But he still wants (b) to defend a version of logicism about the natural numbers (using introduction and elimination rules in a first-order context). So how does he square his ambition (b) with his vivid recognition that (a)?

Tennant  writes:

Logicism maintains that Logic (in some suitably general and powerful sense that will have to be defined) is capable of furnishing definitions of the primitive concepts of this main branch of mathematics. These definitions allow one to derive the mathematician’s ‘first principles’ of number theory as results within logic itself. The logicist is therefore purporting to uncover a deeper source of justification for these ‘first principles’ than just that they seem obvious or self-evident to mathematicians working in the branch of mathematics in question, … [p. 5]

So he is out to defend what Wright 2020 calls the “Core logicist thesis” that at least we can get to the mathematician’s familiar Peano Axioms starting from logic-plus-definitions.  And the methods of his version of logicism, Tennant says,

can be used to determine to what extent the truths in a particular branch of mathematics might be logical in their provenance. So it is more nuanced and discerning than logicism in its original and ambitious form, even when confined to number theory. [pp 5–6]

the formulation of mathematical theories in terms of introduction and elimination rules for the main logico-mathematical operators furnishe[s] a principled basis for drawing an analytic/synthetic distinction within those mathematical theories. [p. 13]

So Tennant is quite happy with an analytic/synthetic distinction being applied within the class of first-order arithmetical truths. In fact, that was already his view back in 1987, when his version of logicism in Anti-Realism and Logic dropped more or less stone-dead from the press (Neil has a terrible habit of trying to take on too much at once, as I’d say he does in his latest book — so that earlier book had something to annoy everyone, and I doubt that very many got to the final chapters!)

Whether Tennant should be happy about the idea of non-logical arithmetical truths concerning logical objects is another question, of course, but that’s the line.

OK: so the criterion of success, for Tennant, is that the logicist

accounts for what kind of things the natural numbers are, and thereby also enables one rigorously (and constructively, and relevantly) to derive as theorems the postulates of ‘pure’ Dedekind–Peano arithmetic, which the pure mathematician takes as first principles for the pure theory of the natural numbers. [p. 51]

NB ‘derive’ — it is syntactic provability as against semantic consequence more generally that is in question.

To be sure, we share Frege’s logicist aspiration to establish at least the natural numbers as logical objects, and to derive the Dedekind–Peano postulates that govern them from a deeper, purely logical foundation. Moreover, we claim to have succeeded where Frege himself had failed, for want of a consistent foundational logical theory. The natural numbers, though sui generis, are logical objects. They are recognizable and identifiable as such because of the role they play in our thoughts about objects that fall under sortal concepts. Our logico-genetic path to the natural numbers has proved to be fully logicist. And it does not take Hume’s Principle as its point of departure. [pp 54–55]

And a bit later — and here we interestingly link up with some remarks of Pregel’s which also touch on what has come to be called “Isaacson’s Thesis” — Tennant writes

Note that it will suffice, for the natural logicist to be able to claim substantial success in this project, to recover the axioms of Dedekind–Peano arithmetic. Indeed, if the natural logicist manages to succeed only on Dedekind–Peano arithmetic, this might offer the explanation sought by Isaacson [1987] of why it is that the axioms of Dedekind–Peano arithmetic are so very ‘natural’. As Isaacson puts it,

… Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. … it consists of those truths which can be perceived directly from the purely arithmetical content of a categorical conceptual analysis of the notion of natural number.

We have already acknowledged Gödelian incompleteness in arithmetic, and we fully recognize the logical and epistemological challenge posed by it. [p. 69]

So, for Tennant it is an open question how far logicist methods will take us into arithmetic. But that doesn’t impugn, he thinks, its success in giving us the mathematician’s ‘first principles’ for arithmetic. And indeed it might be conceptually important that the logicist method takes us just so far into arithmetic and no further, in the spirit of something like Isaacson’s ideas.

Pregel very reasonably asks the Wright-style neo-logicist

In particular, what account is the Neo-Logicist to offer of the analyticity status of the semantic consequences of HP that are not deductive consequences? Are they analytic as well, though for a different reason? Or synthetic? And how do we account for the fact that different possible choices of second-order deductive systems mean different formulas get categorised as ‘core’?

Tennant would bite the bullet for his version of logicism — they’re synthetic. And his logical framework is first order, so he doesn’t hit the second problem.

OK, that’s all mostly tangential to Pregel’s concerns with the canonical version of neo-logicism. But the point at which they both touch on Isaacson is interesting and suggestive — though since Tennant isn’t tangling with second-order logic, he avoids some of the worries Pregel rightly raise about whether we can deploy something like  Isaacson’s Thesis in the canonical second order framework.

Now, I hasten to add, all this isn’t to say that I outright endorse Tennant’s version of logicism. But do I find myself suspecting that his way with Gödel by limiting the ambitions of a logicism is a “best buy” for someone who wants to rescue something of substantive interest out of a latter-day version of logicism.

And now I’m kicking myself — I have just remembered that I meant to change a footnote about neo-logicism in Gödel Without (Too Many) Tears having read Neil when his book came out, and I forgot to do so before publishing the second edition last month. Bother!

Adrian Moore, on Gödel’s Theorem, briefly

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There has just been published another in the often splendid OUP series of “Very Short Introductions”: this time, it’s the Oxford philosopher Adrian Moore, writing on Gödel’s Theorem. I thought I should take a look.

This little book is not aimed at the likely readers of this blog. But you could safely place it in the hands of a bright high-school maths student, or a not-very-logically-ept philosophy undergraduate, and they should find it intriguing and probably reasonably accessible, and they won’t be led (too far) astray. Which is a lot more than can be said for some other attempts to present the incompleteness theorems to a general reader.

I do like the way that Moore sets things up at the beginning of the book, explaining in a general way what a version of Gödel’s (first) theorem shows and why it matters — and, equally importantly, fending off some initial misunderstandings.

Then I very much like the way that Moore first gives the proof that he and I both learnt very long since from Timothy Smiley, where you show that (1)  a consistent, negation-complete, effectively axiomatized theory is decidable, and (2) a consistent, sufficiently strong, effectively axiomatized theory is not decidable, and conclude (3) a consistent, sufficiently strong, effectively axiomatized theory can’t be complete. Here, being “sufficiently strong” is a matter of the theory’s proving enough arithmetic (being able to evaluate computable functions). Moore also gives the close relation of this proof which, instead of applying to theories which prove enough (a syntactic condition), applies to theories which express enough arithmetical truths (a semantic condition). That’s really nice. I only presented the syntactic version early in IGT and GWT and (given that I elsewhere stress that proofs of incompleteness come in two flavours, depending on whether we make semantic or proof-theoretic assumptions) maybe I should have explicitly spelt out the semantic version too.

Moore then goes on to outline a proof involving the Gödelian construction of a sentence for PA which “says” it is unprovable in PA, and then generalizes from PA. (Oddly, he starts by remarking that “the main proof in  Gödel’s article … showed that no theory can be sufficiently strong, sound, complete and axiomatizable”, which is misleading as a summary because Gödel in 1931 didn’t have the notion of sufficient strength available, and arguably also misleading about the role of semantics, even granted the link between \Sigma_1-soundness and \omega-consistency, given the importance that Gödel attached to avoiding dependence on semantic notions. The following text does better than the headline remark.) Moore then explains the second theorem clearly enough.

The last part of the book touches on some more philosophical reflections. Moore briefly discusses Hilbert’s Programme (I’m not sure he has the measure of this) and the Lucas-Penrose argument (perhaps forgivably pretty unclear); and the book finishes with some rather limply Wittgensteinean remarks about how we understand arithmetic despite the lack of a complete axiomatization. But I suppose that if these sections spur the intended reader to get puzzled and interested in the topics, they will have served a good purpose.

My main trouble with the book, however, is with Moore’s presentational style when it comes to the core technicalities. To my mind, he doesn’t really have the gift for mathematical exposition. Yes, all credit for trying to get over the key ideas in a non-scary way. But I, for one, find his somewhat conversational mode of proceeding doesn’t work that well. I do suspect that, for many, something a bit closer to a more conventionally crisp mathematical mode of presentation at the crucial stages, nicely glossed with accompanying explanations, would actually ease the way to greater understanding. Though don’t let that judgement stop you trying the book out on some suitable potential reader, next time you are asked what logicians get up to!

Gödel Without (Too Many) Tears — 2nd edition published!

Gödel's theorems, Logic / 1 Comment /

Good news! The second edition of GWT is available as a (free) PDF download. This new edition is revised throughout, and is (I think!) a significant improvement on the first edition which I put together quite quickly as occupational therapy while the pandemic dragged on.

In fact, the PDF has been available for a week or so. But it is much nicer to read GWT as a physical book (surely!), and I held off making a splash about the finalised new edition until today, when it also becomes available as a large-format 154pp. paperback from Amazon. You can get it  at the extortionate price of £4.50 UK, $6.00 US — and it should be €5 or so on various EU Amazons very shortly, and similar prices elsewhere. Obviously the royalties are going to make my fortune. ISBN 1916906354.

The paperback is Amazon-only, as they offer by far the most convenient for me and the cheapest for you print-on-demand service. A more widely distributed hardback for libraries (and for the discerning reader who wants a classier copy) will be published on 1 December and can already be ordered at £15.00, $17.50. ISBN: ‎ 1916906346. Do please remember to request a copy for your university library: since GWT is published by Logic Matters and not by a university press, your librarian won’t get to hear of it through the usual marketing routes.

Right. And now to get back to other projects …

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

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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

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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

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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

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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

Gödel's theorems, Logic / 2 Comments /

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 …

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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

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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!

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