Here, then, is a full draft of Part I of the revised *Beginning Mathematical Logic* Study Guide (vi + 114pp).

It is, I hope, in a reasonable state. But it goes without saying that comments, suggestions and corrections of the inevitable typos will still be most welcome.

The plan now is to add the much shorter Part II (briefly looking over the fence at some areas of logic mostly of interest to philosophers rather than mathematicians), and then the more substantial Part III (advanced readings on the topics of Part I). But hopefully, revising these further chapters will be relatively speedy, as I won’t be writing expansive overviews for them. I want this time-consuming project off my laptop by the end of the year!

Rowsety MoidThe Guide is very much the sort of thing I like, and I look forward to reading this version properly.

Some thoughts after an initial scan:

1. It would be useful to have a date or version indication of some sort in the text, so that people will know which version they have and can refer to it when quoting or making comments. (I’ll call this one the version of 4 November 2021.)

2. As I might have mentioned earlier, Ebbinghaus, Flum, and Thomas is now in a 3rd edition.

3. I like having the index of authors.

4. What do you think of Torkel Franzén’s

Inexhaustibility: A Non-Exhaustive Treatment(which isn’t mentioned in the Guide)? I thought it excellent the last time I had a good look (which was admittedly some years back). It takes a while to reach the 1st incompleteness theorem (p 216), but that is because it covers many things along the way, including ordinals and ordinal notations (Gentzen’s consistency proof is mentioned, though not given, on p 202), and sets things up using deducibility conditions so that the 2nd incompleteness theorem can appear shortly after (p 217), followed by Löb’s theorem, before going onward (in the three concluding chapters) towards inexhaustibility. The book has illuminating discussions throughout.5. I like seeing Kleene’s

Mathematical Logicmentioned. My interest in logic began when I found a copy of that book in the school library.6. I think the placing and description of Halmos,

Naive Set Theoryis good, given that it pretty much has to be mentioned.7. I think your description of George Tourlakis’s

Set Theorycould give readers a misleading impression. It’s true that he doesn’t present Cantor’s uncountability theorem until p 455. However, that’s largely because of the topic order he’s chosen — by then he’s done ordinals and even the constructable universe — and because there’s been a lot of motivational and explanatory discussion along the way. Not because of excessive formalism (though there is some of that).I don’t think Tourlakis is the best choice as a first set theory book, but I would also say that about some of the other books. Instead, its role is more as a supplement or a second book, to overlap, go further, and take a different approach (as in the strategy described in section 1.6). I think its coverage of more advanced topics is what makes it interesting and useful. In your description, it’s not even clear that it covers them. (You do mention the chapter on forcing, but

a chapter on forcingcould just be a high-level overview.)On excessive formalism, it’s true that Tourlakis uses formal proofs more than other books — I think that’s largely what the ‘excessive formalism’ amounts to — but that’s mostly in the early parts of the book. He then moves to more informal proofs. Most books expect readers to take it on faith that proofs can be formalised. Tourlakis doesn’t, quite, and I think that pays off, to an extent, at some points later on; but I agree many could find it off-putting. If Tourlakis is their 2nd (or later) set theory book, however, they can go quickly through those parts.

One of my ‘tests’ for set theory books is what they say about cofinality, if they cover it at all. Compare Tourlakis with, for example, Devlin. I don’t see excessive formalism in Tourlakis there. Devlin’s more formal. Devlin gives a formal definition and then starts using it. Unlike in Tourlakis, there’s little, if any, attempt to motivate the concept or to give the reader an intuitive feel for it. Many books are like Devlin’s when they get to such topics. (Enderton is an exception, but he just barely gets to cofinality at the end.)

Peter Smith1. Good point.

2. Oh yes, you did, thanks!

3. Good.

4. Love the book, it is mentioned in Part III of the Guide (but when I come to revise that, I’ll see if the book could/should already be mentioned in a further readings section in Part I

5, 6. Good!

7. It’s quite a while since I looked at Tourlakis on set theory. Perhaps I’ll take another look when I have a chance. (Others too have spoken well of it.)

Aron TProfessor Smith

First thank you for this website and all the amazing resources you provide here. I first got interested in logic and mathematical foundations over 50 years ago when I was in high school. Copi was the first book I read, followed by Langer. I had my parents buy me Principia Mathematica for my 16th birthday (the full 3 volumes) but got stuck on the theory of types. I did manage to complete Quine’s Mathematical Logic and then left logic behind when I started university. A major in philosophy and academic life lost its appeal. Still, foundational Mathematical Logic has not totally abandoned me. I work in software technology and am greatly amused how Russell’s theory of types has found new relevance in computer science.

Recently, being “semi retired” I decided to take up Logic again and “do it right”, starting from the very beginning. I stumbled on your site and an earlier version of your guide. It’s as if I have found a most excellent private tutor to guide me on my path. I waited eagerly for the release of the second version of IFL to start this journey and bought the ebook when it came out.

A couple of months ago I finally started my reading. While the study guide is my path for the future (and I do feel your revision is an improvement, particularly for me) my most useful resource as I read your book, is your Logic Bites. They are like a lively lecture that motivates the reading and provides important insights. Which leads me to the point of this comment: do you have any time line or plan to finish those in the foreseeable future? If yes, I’ll likely wait to complete your text till those are finished and move on to other readings in the interim. If not, I’ll move forward in IFL till the end and go from there.

Again I can’t say enough what an amazing resource you have created and how grateful I am for your work. It is a beautiful achievement and legacy of your professional accomplishments. If only retired professors in other fields would be as generous with their time and sharing their accumulated wisdom. Many, many thanks.

Peter SmithThanks for your kind remarks!

As to whether there will be more “logic bites” — I dashed off the existing ones pretty quickly, and then paused to see whether they actually get read enough to make it worth the effort to revise them, and then continue to write more.

And — bother! — downloads have been neither so few as to make it clear it isn’t worth continuing with them, nor so many as to make me rush back to work on them …

I think I probably will return to them in the new year at some point: but please don’t hold your breath!