I’m almost done with the category theory book — at least as far as the draft content is concerned. There will be tinkering to be done (in particular with respect to having a consistent line on issues of “size”), and then there will be the joys of proof-reading and indexing. But I’m beginning to think occasionally about the next logical project — a new edition of the much-downloaded *Study Guide*. In particular, I’ll want to rethink the early chapters introducing FOL and then making revising suggestions. So what new books have been published over the last few years whose coverage of FOL (and entry-level model theory, in so far as that is distinct) might be usefully recommended?

David Marker (the author of that modern classic *Model Theory: An Introduction*) has just published *An Invitation to Mathematical Logic* covering FOL, some model theory, some computability theory, and making connections with other areas of mathematics. At a quick first look, it is indeed a graduate-level book, not so much an invitation to newbies as something to tackle after entry-level texts.

The next book is by Ieke Moerdijk and Jaap van Oosten (who I think of as category theorists). *Sets, Models and Proofs* was published at the end of 2018, and I’m not sure how I missed it before. It is undergraduate-friendly and very short, only 130 pages before the end matter. And yes, it talks about sets and models before proofs. But, again at a first browse, this looks useful

Dag Westerståhl published his *Foundations of Logic: Completeness, Incompleteness, Computability* at the beginning of this year. Evidently a retirement project, based on a long teaching history. I’ve read some with enjoyment, and I do like the tone a lot. I think that at least parts of this could well be entry-level recommendations. So high on my list of readings for when I seriously get down to revising the *Study Guide*.

*Three Views of Logic* by

Then the last part of the book, perhaps misleadingly entitled “Philosophical Logic” is about non-classical logics, particularly for relevance systems. Not my cup of tea, but looks well done.

Lastly for the moment, I’ll certainly want to consider *Gödel’s Theorems and Zermelo’s Axioms* by Lorenz Halbeisen and Regal Krapf, published four year ago. Part I introduces FOL. Part II is on Gödel’s Completeness Theorem, and a little more. Part III is about Peano Arithmetic and Gödel’s two Incompleteness Theorems (and proves the completeness of Presburger arithmetic too). Then Part IV, which is titled ‘The Axiom System ZFC’ is in fact about quite a bit more, including models of set theory, the role of ultraproducts, about non-standard models of PA and of the reals, and more.

This is a relatively short book, just 234 pages, and that wide-ranging menu will tell you that the book has to proceed at some pace. But my initial sense when I looked at the book when I first bought it is that the level of exposition is very clear, terse but with enough motivational pointers, and could well please some readers.

So there’s some homework for me, to fill idle hours, taking more careful looks at these books. But what else have I missed, of recent — or perhaps not-so-recent — books, particularly on FOL, that could be worth recommending in the *Guide*?

I don’t trust the absolute download stats, but relative numbers must be reliable. As I’ve said before, the Guide is by far the most downloaded of the *Big Red Logic Books *(as many times a month as *IFL* and *IGT* together), and the number of downloads increases over time. And to my continuing surprise, it actually sells about 600 paperbacks a year too. So it must be worth putting in the effort to update/improve it. And, sad to relate, I do rather enjoy working on it …

T. S. RobinsonThough not new (dating from 2009), they seem to have been overlooked: Nuel Belnap’s

Notes on the Art of Logic(https://sites.pitt.edu/~belnap/nal.pdf) andNotes on the Science of Logic(https://sites.pitt.edu/~belnap/nsl.pdf).These texts are free to access and download via Belnap’s Pitt faculty page.

They cover FOL in a nice pedagogical way that should make them attractive to independent learners. Belnap is quite patient in explaining everything presented, with plenty of pointers and casual asides to keep the technical aspects wholly lucid for students.

Considering that they’re free, readily available online, and of good exegetical quality for self-learners, I thought I might flag them here as potentially good choices to consider for inclusion among the Guide’s “basic recommendations” or “parallel readings.”

Rowsety MoidUnfortunately, most of those books are problematic from the point of view of buying a copy. Only

Sets, Models and Proofsis currently available from Amazon UK. It looks likeThree Views of Logicmay be out of print. It’s too soon forFoundations of Logic: Completeness, Incompleteness, Computabilityto be out of print, but Amazon doesn’t offer it, and though it’s available from third-party sellers, it’s much too expensive for an 80 page book (over £34). Marker’sAn Invitation to Mathematical Logicis also available only from third-party sellers, and their expected delivery dates suggest that they do not actually have any copies. I expect it will eventually become easier to buy, but right now it’s not clear it’s actually been released despite the supposed release date of 7 May 2024.*

A new book that’s possibly of interest is

Intuitive Axiomatic Set Theoryby José L Garciá.Chris GrantMarker’s book is available right now to those fortunate enough to be affiliated with an institution with a Springer site license.

Peter SmithThree Views of Logicremains available from Princeton, though a pricey paperback. Still, libraries should be encouraged to get it. Marker’sInvitation to Mathematical Logic, like most new Springer books, seems to be print-on-demand, and is available from Amazon or direct from Springer. But as Chris notes, it should be e-available in major libraries via their deals with Springer. (Those two, likeSets, Models and Proofsare also locatable as PDF downloads, between you and your conscience.)Westerståhl’s

Foundations of Logic: Completeness, Incompleteness, Computabilityis available from various sellers linked on Amazon: I got one a couple of months back. it is in fact 451pp.Garcia’s book is on my radar for when I get round to the set theory chapters!

PeterGödel’s Theorems and Zermelo’s Axioms

https://library.lol/main/D6ABA45C3E95B0D85E9DC705566C27B2

Peter SmithOops, senior moment, it’s on the bookshelf behind me as I type. I’ll add this to the post!

Peter FI am helping to prepare a class on intermediate logic for next year and I wanted to work through Marker’s book for practice problem inspiration. Unfortunately, there are a serious mass of typos starting from the first few pages and going on through the entire book. Some of them are minor like repeated words, but some are worse (when setting up Beth’s definability theorem as an exercise, the definitions for explicit definability and implicit definability are both called explicit and I’m not sure someone who is studying on their own would be able to figure out which one is supposed to be implicit just by the surrounding text). Although I do like the writing style, the coverage of advanced topics, and the overall organization, I can’t get over how many mistakes there are as they become distracting very quickly.

Karo MaestroThis is not an uncommon problem with first editions of textbooks, Peter. Especially these days when the monopoly textbook publishers like Springer ride the backs of authors to get the book in print and the cash flowing ASAP.

They don’t really proofread them carefully much anymore-especially with books by popular authors like Marker. You can help with this by making a detailed error list and emailing it to the author or Springer. And if you don’t have time, suggest your students do it while recommending the book!

Lim Hyun MinMaybe Kenneth Kunen’s

Foundations of Mathematics?Peter SmithGood thought: I should revisit this book which, indeed, is not mentioned in the current

Guide.Karo MaestroYou ABSOLUTELY should include both of Kunen’s recent graduate level logic books in a new edition of the Guide, Doctor Smith.

The Foundations book by Kunen is very terse-it’s clearly for advanced mathematics students. I think even strong first year graduate students in mathematics will find it difficult. It’s really hard to imagine graduate students in philosophy being able to work through it unless have a comparable background in mathematics to go with their philosophy. The second edition of his classic Set Theory textbook is just slightly more accessible. As good as both books are, I don’t know if they’re suitable for an advanced philosophy audience. They have to be mathematically literate and willing to bring a stack of scrap paper and a pen to literally chew through every line of each book.

I’m glad you noticed Marker’s new text. I’ve glanced it over-it’s also clearly not a book for philosophers. It’s a text for first or second year graduate students in mathematics. Which is a great thing since there just aren’t many choices for such textbooks. Most of the content of the book is in the exercises. Despite it’s difficulty level, Marker writes very well with enormous clarity. I think graduate students in both mathematics and philosophy would benefit enormously from working through it. I also agree with Marker that axiomatic set theory shouldn’t be covered in a logic book-it’s vast enough of a subject that really needs it’s own separate treatment as a follow-up after a basic mathematical logic course.

But the main part of the Guide that needs to be completely updated and expanded is the section at the end on type theory. Type theory has become far more then just a curious anachronism of the days before modern set theory. It’s now a deep and important area that connects mathematical logic, philosophy of mathematics and computer science in a way it’s founders could have never anticipated. The book by Andrews is beautifully written and challenging-it’s really the only type theory text for modern mathematicians.I think you should take a deeper look at it. Also, William Farmer-who’s wonderful paper, “Seven Virtues of Simple Type Theory”, is deservedly becoming a modern classic-has written a full book length treatment of type theory based on the type system developed in Andrews. The book is SIMPLE TYPE THEORY. I’ve only glanced through it, but it seems very accessible-and that’s needed.

Peter SmithThanks for this. (The revised Kunen set theory book is already, very briefly, mentioned in the

Guide— I still suspect the original version is more accessible of self-study. )I agree completely about needing to radically revise the current too-brief remarks on type theory, and indeed recently got a copy of Farmer’s book, which will certainly get a recommendation.

Karo MaestroI have one more book to recommend you review for the second edition of the Beginner’s Guide-and frankly, it’s absence completely mystifies me. It mystifies me because it’s the only comprehensive mathematical logic book I regularly recommend to graduate students for self study despite it’s age. Most are struggling with Shoenfield and are completely unaware of this book.

Why isn’t Machover and Bell’s A COURSE IN MATHEMATICAL LOGIC included in the Guide? I looked twice and still don’t understand why it’s missing.

Peter SmithBell and Machover is, as it happens, on my list of books to revisit (I think, if I recall rightly, that it was recommended in a much earlier iteration of the

Guide). And certainly I’d recommended it over Shoenefeld for self-study.I’m not too convinced, however, about the virtues of “comprehensive mathematical logic books” as a genre, as opposed to using different first-choice books on core FOL, entry-level model theory, entry-level computability theory, etc.

PeterIterative Conceptions of Set

by Neil Barton has been published in the insanely expensive Cambridge intro series?

Can you recommend it or even justify its price?

PeterHi!

Sorry to bother you again but I just found mentioned book and read the first pages. I am still not firm in logic and I came across a typo and mistake (but I am not sure, that is why I would like to know your opinion.

Here is the link: http://library.lol/main/16A94ACAFDAFE411348CF9CF967F192C

And here is my question:

On page 18 Barton explains Cantor’s paradox regarding the universal set. He writes:

“Consider the condition x = x. Let {x|x = x} be denoted

by u (for “universal set”). Now consider P(u), namely the powerset of u.

By Naive Comprehension, this is also a set. Now we show x = P(u) by

noting: (i) every element of P(u) is an element of u (trivially), and (ii) if

x ∈ u, then x ∈ P(u) (since if x ∈ u, then ∀y ∈ x, y ∈ u (i.e. x ⊆ u) and so

x ∈ P(u)). So, u = P(u).

Clearly then, there is a surjection f : u ↠ P(u). Now consider the set

c = {x|x < f(x)}. Since f is surjective, there is a y ∈ u such that f(y) = c.

We now ask “Is y ∈ c?” If yes (i.e. y ∈ c), then y ∈ f(y), but then y violates

c’s defining condition, and so y < c, contradiction. So then we assume

y < c. But then y < f(y), and so y meets c’s defining condition, and y ∈ c,

contradiction. So y ∈ c ↔ y < c, a contradiction!

In fact, this proof can be transformed into a proof of Cantor’s theorem,

just by replacing u by any old set x and performing a reductio on the claim

that there is a surjection f : x ↠ P(x).”

My issues regard the first paragraph:

Should the fourth sentence not start with “Now we show u = P(u)” instead of “Now we show x = P(u)”? Furthermore, as far as I understand the universal set it contains everything, not just all sets but also things that cannot be reduced to sets (say chairs). But if that is the case, the proof in the first paragraph just does not work in my opinion. i) of the proof is trivial, I agree. But ii) presupposes that any element x of the universal set u has itself members and therefore is a set. Initially I thought plain natural numbers (also included in u) would count as counterexamples as well but I guess it is assumed that those are to be reduced to sets. Apart from my worries about the correctness of the proof. What is u usually thought to contain? Just sets and numbers thought as sets?

PeterOne last post! Apologies for spamming. Please ignore my posts or delete them. I asked my question on stack exchange:

https://math.stackexchange.com/questions/4948120/cantors-paradox-and-the-universal-set?noredirect=1#comment10581411_4948120