# A First Journey Through Logic?

Home from Rhodes, post-flight covid tests thankfully negative, a few domestic chores done, and so it’s back to work on the Study Guide …

The initial task is to tidy the hundred or so pages of Part I of the Guide, covering the core mathematical logic curriculum at an elementary level. I’ve posted drafts of the nine  chapters here on the blog, which have each been downloaded hundreds of times. But I’ve had almost no comments/correctiond. Which I’ve decided to take in a cheerfully positive spirit — I like to infer that people don’t think I’m going so horribly wrong that they need to protest loudly (if only to stop me leading their students astray)! So the tidying work will, I hope, mainly be a matter of trying to imposing greater consistency of level and tone between the nine chapters.

However, first, I had better do a bit of homework, and get acquainted with a fairly recent book which I hadn’t noticed before, A First Journey through Logic by Martin Hills and François Loeser, published in the AMS Student Mathematical Library in 2019. “The book starts with a presentation of naive set theory, the theory of sets that mathematicians use on a daily basis. Each subsequent chapter presents one of the main areas of mathematical logic: ﬁrst order logic and formal proofs, model theory, recursion theory, Gödel’s incompleteness theorem, and, ﬁnally, the axiomatic set theory. Each chapter includes several interesting highlights— outside of logic when possible either in the main text, or as exercises or appendices.” Which is a promising prospectus, with its chapters covering just the same main topics as Chapters 2 to 7 of the Study Guide. However, this is a book of under two hundred small-format pages. Which rather suggests that either the authors aren’t aiming to get very far on any particular topic (though such a book can be very useful when well done, cf. Robert Wolf’s A Tour Through Mathemtical Logic). Or else our authors must have written with considerable compression at the probable cost of ready accessibility.

I’m afraid that our authors have taken the second line. They say in their Introduction that the book originates from a course taught at École Normale Supérieure: and it reads exactly like souped-up lecture notes spelling out carefully lots of technical details, to back up a course of lectures which explain the rationales for the various formal constructions. But without the explanatory, arm-waving, motivational chat, the arid formal details make for a hard and uninviting journey. Perhaps some chapters might serve as brisk revision material: but surely they not the place to make a beginning on mathematical logic.

I’ll say just a bit more in two follow-up posts.

### 4 thoughts on “A First Journey Through Logic?”

1. The AMS Student Mathematical Library is a peculiar thing. The books don’t seem especially good; they do seem especially expensive. This one, for instance, is RRP £54.50 for a 195 page paperback.

In any case, I think the title — A First Journey through Logic — is misleading and makes it look much more introductory than it is. As its Introduction explains, it’s “intended towards advanced undergraduate students, graduate students at any stage, or working mathematicians”.

Anyway, I’ll be interested in what you think of its proof of the 2nd incompleteness theorem.

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I’ve been meaning to comment on the Logic Guide but haven’t managed to find the time. One thing I remember from when I was looking at it earlier is that, in the FOL chapter, it says Logical Options is a “a potential alternative to Bostock at about the same level”. I think it’s a different type of book that covers very different things. It’s more like an alternative to Graham Priest’s Introduction to Non-Classical Logic.

1. Yes, the price is crazy for the sort of book it is intended to be.

Yes, indeed the authors make clear it is for advanced undergrads upwards — though also that it is for those “who seek a first exposure to core material of mathematical logic”. So it is supposed to be, shall we say, fast-track introductory. How well do that succeed at that mission? Watch this space!

Yes, that remark about Bell, DeVidi and Solomon was misleading (and I had indeed recently edited it out) — I think I’d meant a comparability of level rather than content, but that isn’t what I’d said!

2. BTW, also about the FOL chapter, you were very positive about Richard Kaye’s The Mathematics of Logic originally, in your old blog. That “Two new logic books ….” post no longer seems to be available except via the Wayback Machine. But it is there:

https://web.archive.org/web/20070902220702/http://logicmatters.blogspot.com/2007/07/two-new-logic-books.html

(The other book is Chiswell and Hodges, Mathematical Logic.)

That post stuck in my mind because it made the books sound so interesting that I then bought a copy of each.

Now, admittedly, your assessment back then was based on “a rapid glance through”, and even now you say Kaye’s book “could be very interesting and illuminating” if “you already know a fair amount of this material from more conventional presentations.” Still, it seems to me that this is one of the books that suffers a bit, in the Guide, from not being the best starting point (or “initial introduction”, “early trip”).

And it’s buried away in an obscurely introduced section: What about Ebbinghaus, Flum and Thomas? Hedman? Hinman? Rautenberg? Kaye?. Will the Guide’s intended readers even know which books that’s about? Of course, they could read on and see. That’s not encourage by the way the section begins, however, and books you say something positive about (such as Kaye’s) are mixed in with ones you don’t.

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There’s a new, 3rd, edition of Ebbinghaus et al.

1. Thanks for this — as it happens I’ve just been looking again at the FOL chapter and was myself struck by the fact that the last section is a bit of a untidy heap of somewhat disparate comments. I haven’t decided quite how to handle this: but I quite agree some major tidying is called for!

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