A very short book note

With a mind to building myself a reading list for homework before updating the BML Study Guide, I have been doing some online searches for recent publications on mathematical logic. I asked ChatGPT and Claude in “research” mode too, but they were pretty useless. However, a title they both noted, new to me, was Mathematical Logic: An Introduction by Daniel Cunningham, published by De Gruyter in 2023 as a quite exorbitantly priced paperback. I’ve taken a look at a copy lurking online.

There are chapters on propositional logic and FOL, with soundness and completeness proofs, and then chapters on computability and on undecidability and incompleteness. So this complements Cunningham’s 2016 book on set theory for CUP.

I do mention that earlier book in the Guide, p. 89. And I’m afraid I summarily dismiss it, with the remark that “Its old-school Definition/Lemma/Theorem/Proof style just doesn’t make for an inviting introduction for self-study.” So I didn’t get my hopes up for this new effort. And I can only report that this really does seem more of the same. I just can’t warm to the rather relentless style, unleavened by enough by way of engaging explanatory classroom chat.

I dipped in and out and repeatedly found what are beautiful and elegant ideas rather submerged under the formalities. And I could argue about some of the expository choices too. For example, having built up all the needed apparatus, the treatment of Gödel’s theorems at the end of the book is perhaps oddly rushed (we get an incompleteness theorem resting on a semantic premiss, but not the syntactic version which usually gets the honorific title of the first incompleteness theorem).

Maybe these books could work as rather austere monochrome notes accompanying a suitably colourful and motivating lecture course (or work as revision materials). But for initial self-study of their topics? Not so much. But judge Cunningham’s style for yourself, in a minor case. There’s a preview of the front matter and the first ten pages here. Ask yourself if it is going to be obvious to the student neophyte what is going on at pp. 8–10(!) about the “free generation” of sets. How easy to follow is that presentation from a standing start? I rest my case.

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