Kurt Gödel: Results on Foundations

If you have access to a library which subscribes to Springer Link, you should be able to download an e-copy of this very recent addition to the growing list of editions of Gödel’s various notebooks. (If you don’t have good library access, then tough — Springer are price-gouging at £111.50 for the PDF, and more for the print-on-demand version.)

The editors Maria Hämeen-Anttila and Jan von Plato write in their short Preface

If there is one “must” to be cleared in the enormous mass of the Kurt Gödel Papers kept at the Firestone Library of Princeton University, it is the series of four notebooks titled Resultate Grundlagen. Gödel wrote these 368 pages between 1940 and 1942, except for the first 33 and last 12 pages. There is a continuous page numbering and the same goes for the theorems. It has been a great fortune for us to meet the task of transcribing, translating, and editing these notebooks.

And later, in their introductory essay

Resultate Grundlagen [RG] is a collection of results Gödel considered finished. … Close to two thirds of RG deal with set theory … Next to set theory, RG contains results on arithmetic and recursive functions. Type theory is one clearly separate topic, and so is what Gödel called “positive logic.” The latter relates to intuitionism which was one of Gödel’s permanent interests from the early 1930s on. This interest is clearly seen in [RG] with about one part in four devoted to intuitionistic logic and its interpretation.

So that tells us two things. First, about the topics of the RG notebooks themselves. And second, inadvertently, that the language of this edition is sometimes only an approximation to good English. Evidently, Springer’s contribution to the publishing of this book didn’t run to a native-speaker copy-editor. This matters, I think, for two reasons. First, readers for whom English is not their first language will stumble. Second, the editors have (oddly to my mind) not given their transcription of Gödel’s obsolete German shorthand in a parallel text (surely an achievement worth preserving for future researchers): so occasionally the reader might wonder whether seemingly odd or stuttering phrasing is in the original or is a result of rendering into clumsy English. In fact the editors write

RG is a polished shorthand text when compared with such sources of preliminary work as [other notebooks]. There are next to no cancellations, but there are additions that often result in awkward sentence structures. The question is to what extent such passages should get improved in translation.

Given this sort of issue, why indeed not pre-empt a reader’s questions with a parallel text, as in the canonical edition of the Works?

On the key set-theoretic content, the editors write

After the transcription and translation work was done, we were lucky to find in Akihiro Kanamori a reader without comparison of Gödel’s results on foundations. … Aki took up the task and presented us with a splendid essay on The remarkable set theory in Gödel’s 1940–42 Resultate Grundlagen, an essay that explains how Gödel had arrived at numerous results independently discovered by others later, sometimes much later, in an anticipation of the development of set theory from 1942 on, the year Gödel left formal work in logic and foundations.

Which is good to know; but since Kanamori’s essay isn’t included in the book as an introduction (and isn’t yet available elsewhere), the rest of us will have to wait a little for a knowledgeable guide to Gödel’s achievement in RG. All that said, it remains astonishing to find how productive Gödel was in those years when he was publishing so little. Fascinating but frustrating to dip into.

Mileti, Modern Mathematical Logic, Chs 7–10

To continue. Chapter 7 of Mileti’s MML is titled “Model theory”. Of the five sections, the first three can’t be recommended. In particular, §7.2 makes such heavy weather of that fun topic, nonstandard models of arithmetic and analysis. There are so many alternative treatments which will be more accessible and give a more intuitive sense of what’s going on. By contrast, I thought §7.4 on quantifier elimination did a better-than-often job at explaining the key ideas and working through examples. §7.5 on algebraically closed fields worked pretty well too.

And now we get two chapters on set theory, together amounting to almost a hundred pages. There’s a major oddity. The phrase “cumulative hierarchy” is never mentioned: nor is there any talk of sets being found at levels indexed by the ordinals. The usual V-shaped diagram of the universe with ordinals running up the spine is nowhere to be seen. I do find this very strange — and not very ‘modern’ either! There are minor oddities too. For example, the usual way of showing that the Cartesian product of A and B (defined as the set of Kuratowski pairs \langle a, b\rangle) is a set according to the ZFC axioms is to use Separation to carve it out of the set \mathcal{P}(\mathcal{P}(A \cup B)) in the obvious way. Mileti instead uses an unobvious construction using Replacement. Why? A reader might well come away from the discussion with the impression that Replacement is required to get Cartesian products and hence all the constructions of relations and functions which depend on that. (I rather suspect that Mileti isn’t much interested in ‘modern’ finer-tuned discussions of what depends on what, such as the question of  which set-theoretic claims really do depend on something as strong as replacement.)

So: Chapter 8, without explicitly mentioning the cumulative hierarchy (let alone the possibility of potentially more natural axiomatisations in terms of levels) gives us ZFC, and the usual sort of story about how to develop arithmetic and analysis in set theory. The mentioned oddities apart it is generally OK: but the recommendations for entry-level set theory in the Beginning Mathematical Logic Guide do the job better and in a friendlier way. However I should mention that, at the end of the chapter, §8.7 on models, sets and classes, does do the job of explaining the role of class talk rather nicely.

Chapter 9 is on ordinals, cardinals, and the axiom of choice; and I thought this chapter worked comparatively well. (Perhaps the perceived unevenness is all in my mind! And I know from my own efforts in writing long-ish books that maintaining a consistent level of approachability, of proportions of helpful less formal chat around the more formal stuff and so on, is difficult. I can only report how I am finding the book — and, as they say, your mileage may vary.)

Finally in this group, Chapter 10 is much shorter, just two sections on “Set-theoretic methods in model theory”. The first, just four pages, is on sizes of models; and then the second is an opaque and to my mind misjudged ten pages on ultraproducts.

And so it goes: as with the earlier chapters, a mixed bag.

Mileti, Modern Mathematical Logic, Chs 1–3

Towards the end of last year, within a week or two of publishing Jeremy Avigad’s Mathematical Logic and Computation (a bumpy ride, but very well worth having), CUP also released another textbook, Joseph Mileti’s Modern Mathematical Logic. I’d earlier seen a substantial set of notes that Mileti had posted online, and (to be frank) wasn’t over-impressed; so I haven’t been rushing to read this. But I thought I would now take a look at the book version, with a view to seeing whether there are any chapters which I’d want to mention or even recommend in the next iteration of the Beginning Mathematical Logic Study Guide.

Level and coverage? MML is announced as aimed at advanced undergraduates or beginning graduates (by US standards, anyway), though the book is distinctly less ambitious than Avigad’s. Mileti says he assumes familiarity with some basic abstract algebra; however, this seems perhaps more needed to best appreciate some illustrative examples rather than as necessary background for grasping core content. The coverage is broadly conventional, starting with basic first-order logic (though with the opposite emphasis to Avigad: there’s no real proof theory). Then there’s a little model theory, entry-level axiomatic set theory, some computability theory, and a treatment of incompleteness. At this point, then, at least just glancing at the table of contents and diving into the first chapters, I’m not at all sure quite what makes this a book on especially modern mathematical logic in either topics or general approach.

I rather liked the tone of the short Introduction; and going through the next couple of chapters, there is friendly signposting and some nice turns of phrase. But …

But Chapter 2, the first substantial chapter, is thirty pages on ‘Induction and Recursion’. We get a pretty dense treatment of what Mileti calls “generating systems”, three different ways of defining the set of generated whatnots, proofs that these definitions come to the same, then a criterion for free generating systems, a proof we can do recursive definitions over the free systems, and so on. This is all done in what strikes me as a rather heavy-handed way which could be pretty off-putting as a way of starting out. Many students, I would have thought, will just feel they have been made to labour unnecessarily hard at this point for small returns. And when the very general apparatus is applied e.g. in the next chapter to prove, e.g., unique parsing results, I don’t think that what we get is more illuminating than a more local argument. (I suppose my pedagogic inclination in such cases is to motivate a general proof idea by proving an interesting local case first and then, at an appropriate point later, saying “Hey, we can generalize …”.) I note, by the way, that by the end of §2.2 the reader is already supposed to know about countable sets and accept without demur that a countable union of countable sets is countable.

Chapter 3, the next fifty pages, is on propositional logic. A minor complaint is that the arrow connective is initially introduce in the preface as meaning “implies” (oh dear), and then we get not a word of discussion of the truth-functional treatment of the connective unless my attention flickered. But my main beef here is on the chosen formal proof system. This is advertised as natural deduction, but it is a sequent system, where on the left of sequents we get sequences rather than sets (why?). And although the rules are set out in a way that would naturally invite tree-shaped proofs, they are actually applied to produce linear proofs (why?). Moreover, the chosen rule-set is not happily motivated. We have disjunctive syllogism rather than a proper vE rule; double negation elimination is called ¬E; removing double negation elimination doesn’t give intuitionistic logic. OK Mileti  isn’t going to be interested in proof theory; but he should at least have chosen a modern(!) proof system with proof-theoretic virtues!

As for completeness, we get the sort of proof that (a) involves building up a maximal consistent set starting from some given wffs by going along looking at every possible wff in turn to see if it can next be chucked into our growing collection while maintaining consistency, rather than the sort of proof that (b) chucks in simpler truth-makers only as needed, Hintikka style. We are not told what might make the Henkin strategy better than the more economical Hintikka one.

To finish on a positive note, perhaps the best/most interesting thing in this chapter is the final section (and the accompanying exercises) on compactness for propositional logic, which gives a nice range of applications.

To be continued

Logical Methods — on modal logic

Moving on through Greg Restall and Shawn Sandefer’s Logical Methods, Part II is on propositional modal logic. So the reader gets to find out e.g. about S4 vs S5 and even hears about actuality operators etc. before ever meeting a quantifier. Not an ordering that many teachers of logic will want to be following. But then, as I have already indicated when discussing Part I on propositional logic, I’m not sure this is really working as the first introduction to logic that it is proclaimed to be (“requires no background in logic”). I won’t bang on about that again. So let’s take Part II as a more or less stand-alone treatment that could perhaps be used for a module on modal logic for philosophers, for those who have already done enough logic. What does it cover? How well does it work?

Part I, recall, takes a proof-theory-first approach; Part II sensibly reverses the order of business. So Chapter 7 on ‘Necessity and Possibility’ is a speedy tour of the Kripke semantics of S5, then S4, then intuitionistic logic. I can’t to be honest say that the initial presentation of S5 semantics is super-clearly done, and the ensuing description of what are in effect unsigned tableaux for systematically searching for counterexamples to S5 validity surely is too brisk (read Graham Priest’s wonderful text on non-classical logics instead). And jumping to the other end of the chapter, there is a significant leap in difficulty (albeit accompanied by a “warning”) when giving proofs of the soundness and completeness of initutionistic logic with respect to Kripke semantics. Rather too much is packed in here to work well, I suspect.

Chapter 8 is a shorter chapter on ‘Actuality and 2D Logic’. Interesting, though again speedy. But for me, the issue arises of whether — if I were giving a course on modal logic for philosophers — I’d want to spend any time on these topics as opposed to touching on the surely more interesting philosophical issues generated by quantified modal logics.

Chapter 9 gives Gentzen-style natural deduction systems for S4 and S5. Which is all technically fine, of course. But I do wonder about how ‘natural’ Gentzen proofs are here, compared with modal logic done Fitch-style. I certainly found the latter easier to motivate in class. So Gentzen-style modal proof systems would not be my go-to choice for a deductive system to introduce to philosophy student. Obviously Restall and Sandefer differ!

Overall, then, I don’t think the presentations will trump the current suggested introductory readings on modal logic in the Study Guide.

Restall & Standefer, Logical Methods

A new introductory logic textbook has just arrived, Greg Restall and Shawn Standefer’s Logical Methods (MIT).

This promises to be an intriguing read. It is announced as “a rigorous but accessible introduction to philosophical logic” — though, perhaps more accurately,  it could be said to be an introduction to some aspects of formal logic that are of particular philosophical interest.

The balance of the book is unusual. The first 113 pages are on propositional logic. There follow 70 pages on (propositional) modal logic — this, no doubt, because of its philosophical interest. Then there are just 44 pages on standard predicate logic, with the book ending with a short coda on quantified modal logic. To be honest, I can’t imagine too many agreeing that this reflects the balance they want in a first logic course.

Proofs are done in Gentzen natural deduction style, and proof-theoretic notions are highlighted early: so we meet e.g. ideas about reduction steps for eliminating detours as early as p. 22, so we hear about normalizing proofs before we get to encounter valuations and truth tables. Another choice that not everyone will want to follow.

However, let’s go with the flow and work with the general approach. Then, on a first browse-and-random-dipping, it does look (as you’d predict) that this is written very attractively, philosophically alert and enviably clear. So I really look forward to reading at least parts of Logical Methods more carefully soon. I’m turning over in my mind ideas for a third edition of IFL and it is always interesting and thought-provoking to see how good authors handle their introductory texts.

Self-publishing and the Big Red Logic Books

One way of increasing the chance of your books actually being read is to make them freely downloadable in some format, while offering inexpensive print-on-demand paperback versions for those who want them. Or at least, that’s a publication model which has worked rather well for me in the last couple of years. Here’s a short report of how things went during 2022, and then just a few general reflections which might (or might not) encourage one or two others to adopt the same model!

As I always say, the absolute download stats are very difficult to interpret, because if you open a PDF in your browser on different days, I assume that this counts as a new download — and I can’t begin to guess the typical number of downloads per individual reader (how many students download-and-save, how many keep revisiting the download page? who knows?). But here is the headline news:

PDF downloadsPaperback sales
Intro Formal Logic112211112
Intro Gödel’s Theorems7432627
Gödel Without Tears4394677
Beginning Mathematical Logic25863493

No doubt, the relative download figures, comparing books and comparing months, are more significant: and these have remained very stable over the year, with about a 10% increase over the previous year.

As for paperback sales of the first three books, these too remain very steady month-by-month, and the figures are very acceptable. So we have proof-of-concept: even if a text is made freely available, enough people prefer to work from a printed text to make it well worthwhile setting up an inexpensively priced paperback. (In addition there’s also a hardback of IFL which sold 150 copies over the year, and a hardback of the first edition of GWT sold 40 copies up to end of October, before being replaced by a new hardback edition.)

The BML Study Guide was newly paperbacked at the beginning of the year, not with any real expectation of significant sales given the rather particular nature of the book. Surprisingly, it is well on course to sell over 500 copies by its first anniversary.

Obviously, an author wants their books to conquer the world — why isn’t just everyone using IFL? —  but actually, I’m pretty content with these statistics.

To repeat what I said when giving an end-of-year report at the beginning of last January, I don’t know what general morals can be drawn from my experiences with these four books. Every book is what it is and not another book, and every author’s situation is what it is.

But providing an open-access PDF plus a very inexpensive but reasonably well produced paperback is obviously a fairly ideal publication model for getting stuff out there. I’d be delighted, and — much more importantly — potential readers will be delighted, if rather more people followed the model.

Yes, to produce a book this way, you need to be able to replicate in-house some of the services provided e.g. by a university press. But volunteer readers — friends, colleagues and students — giving comments and helping you to spot typos will (if there is a reasonable handful of them) probably do at least as good a job as paid publisher’s readers, in my experience. Writers of logic-related books, at any rate, should be familiar enough with LaTeX to be able to do a decent typographical job (various presses make their LaTeX templates freely available — you can start from one of those if you don’t feel like wrangling with the memoir class to design a book from scratch). Setting up Amazon print-on-demand is a doddle. You’ll need somehow to do your own publicity. But none of these should be beyond the wit of most of us!

The major downside of do-it-yourself publishing, of course, is that you don’t get the very significant reputational brownie points that accrue from publication by a good university press. And we can’t get away from it: job-prospects and promotions can turn on such things. So they will matter a great deal in early or mid career.

But for those who are well established and nearer the end of their careers, or for the idle retired among us … well, you might well pause to wonder a moment about the point of publishing a monograph with OUP or CUP (say) for £80, when you could spread the word to very many more readers by self-publishing. It seems even more pointless to publish a student-orientated book of one kind or another at an unaffordable price. So I can only warmly encourage you to explore the do-it-yourself route. (I’m always happy to respond to e-mailed queries about the process.)

Finally, I can somewhat shamefacedly add a last row to the table above, about work in (stuttering) progress towards an announced but as yet far from finished paperback:

PDF downloadsPaperback sales
Beginning Category Theory7482N/A

This download figure is embarrassing because, as I’ve said before, I know full well these notes are in a really rackety state. But I can’t bring myself to abandon them. So my logical New Year’s resolution is to spend the first six weeks of the year getting at least Part I of these notes (about what happens inside categories) into a much better shape. I just need to really settle at last to the task and not allow myself so many distractions. Promises, promises. Watch this space.

Music for the end of the year

Somewhat to my surprise, I have posted here over a hundred times in the last year. But very many of the posts were of (at best!) pretty ephemeral interest — for example, giving links to then current drafts of the Beginning Mathematical Logic Study Guide, to updated chapters for Gödel Without (Too Many) Tears (lots of those), and updated chapters for the stuttering notes on Category Theory (lots of those too). Other posts were logic/maths booknotes, not all exactly friendly. But I wasn’t always mean: there was warmer praise for a number of books, including the following very mixed bag:

I’ve also been intrigued by the opening chapters of

but I’m not at all sure what to make of Tennant’s deviant form of logicism and his handling of logical objects more generally. And as you’ve noticed, I’m still wrestling with and learning from

More, no doubt, about this very substantial book in the new year.

There have also been a dozen and a half posts on particular musical enthusiasms. So, since you may have a little more time over this holiday season, let me repost links to three wonderful filmed performances which are still available to watch. First, the wondrous Pavel Haas Quartet, recently at Wigmore Hall:

As I wrote before, it makes for a rather dramatic stage presence, Veronika Jarůšková with her mass of golden hair and a golden yellow dress catching the stage lights,  the rest of the quartet in the most subdued of subfusc. And there’s a lot of drama in the performances too. But in one respect, the way the quartet play couldn’t be further from what is visually suggested — the equal balance, the closeness of the ensemble, the intense way they listen to each other, is as ever remarkable. So here they are, playing Haydn’s Op. 76 No. 1, Prokofiev’s second String Quartet No. 2, and then Pavel Haas’s String Quartet No. 2 (that’s the one with percussion in the final movement). On this occasion, I thought, the Prokofiev was especially fine: it is difficult to imagine the deeply affecting Adagio being played better.

Next, here is Elisabeth Brauss, also performing at Wigmore Hall to the warmest of receptions:

The recital started Beethoven’s Op. 109 Sonata, which inspired Elisabeth to quite mesmerising playing with heart-stopping moments: transcendental music, and a performance to more than stand comparison with the very best I’ve heard. Sadly, this part of the recital is no longer available online. But in the rest of an engagingly varied programme she offered us some rarely performed Hindemith, Brahms’ late four Klavierstücke, and Schumann’s Faschingsschwank aus Wien, all done with such verve and then wonderful delicacy, as variously called for — just a delight. You can watch here.

Thirdly, let’s revisit the extraordinarily stellar Lea Desandre, filmed by candlelight, from Rouen … escaping our mad world for an hour. Sheer delight again, and just wonderful singing and playing.

Logicisms and Gödel’s Theorem

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

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!

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 …

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