I have uploaded a half-year “maintenance upgrade” to the Teach Yourself Logic Guide. There are just a few additional entries, a few changes in existing entries, minor re-writing here and there, and then some re-arrangement of material to make it easier for the two different readerships — philosophers and mathematicians — to navigate the beginning of the Guide.
I really must give this project some more love and attention over the coming months (it’s been rather neglected as I have been concentrating on IFL2). In the first half of this year the Guide was downloaded over 15K times from this site, and looked at another 5K times on my academia page (I do still find those stats rather startling). So obviously — patchy and half-baked though it is in many places — there remains some real need for such a Guide. So I guess I should do my best to make it as good as I can. And it is fun enough to work on when in the right frame of mind.
Any suggestions for improvement are of course always welcome! I surely must have missed some recent texts which might be worth looking at. Though I do suspect that the culture of “research assessments” over recent years — where writing such books (as opposed to papers read by eleven people) can count for so very little — puts many people off devoting their energies to writing introductory or mid-level texts.
5 thoughts on “Update to the TYL Guide”
I’ve been thinking lately about the best way to approach model theory, especially beyond the basics (compactness, L-S, etc.). It seems to me that at this level, in order to understand what is going on and have a better motivation for the concepts introduced, one must have a sense of the purpose of it all. Now, I may be biased here by my recent readings, but I’d say that the single program that has driven most of the major research in model theory in the last decades is Shelah’s classification program. Of course, it would be insane to recommend, say, Shelah’s Classification Theory for a neophyte (and even for an advanced learner!). Still, the basic idea behind the classification program is easily explainable (I found the discussion in Button & Walsh, for instance, to be excellent): we want a set of criteria that will tell us whether or not a given theory is tamable. And the paradigm of a tamable theory here are the uncountable categorical theories, as per Morley’s theorem.
I think this is important because:
(1) it shows the importance of Morley’s theorem (I think a nice goal for one who wants to delve into model theory beyond the basics is this theorem);
(2) it highlights the importance of types and “counting types” arguments, since these basically control the models of a theory;
(3) more generally, it provides much of the motivation behind the more esoteric concepts (why I’m studying indiscernible sequences? why is the omitting types theorem important? etc.).
I mention all this because if one quickly reads your guide and proceeds to your suggestions (especially, say, Chang & Keisler, which is an otherwise wonderful book), one may easily be misled into thinking that model theory is a hodgepodge of techniques and theorems that offer no deep insight into the objects being studied. So having something like a program in sight may be useful in correcting this impression.
One other thing: a couple of years ago I commented on the lack of a point of entry into algebra for beginners. I think I now found it: Nathan Carter’s Visual Group Theory. It is extremely elementary and the first three chapters might put a more advanced reader a bit off, but starting with the fourth chapter the story really picks up and I thought it offered some amazing insights into group theory. Since group theory already provides many examples of the algebraic mindset (e.g. isomorphism theorems), I think this is a good introduction for someone with almost no mathematical background (but a certain maturity due to having studied logic). You may want to take a look at it.
Thanks for this! A very good point about avoiding giving the impression that “model theory is a hodgepodge of techniques and theorems”. I’ll put some work into revising this part of the Guide for its next iteration, bearing this in mind.
I’ve just looked at Nathan Carter’s Visual Group Theory, and actually dived into the last couple of chapters, which are indeed very well done. (Why on earth the MAA should publish what is advertised as a student resource so expensively beats me!)
I am a big fan of your guide and also of your book on Godel’s theorems. Your guide does not mention Rautenberg’s A concise introduction to mathematical logic.
Hi Prof. Smith,
I know your guide mainly sticks to published books, but I recently came across this unpublished book by Robert Van Wesep that covers some advanced topics in set theory:
The good news is that the book aims to be self-contained and accessible to a “dedicated amateur.” The bad news, in my opinion, is that the author adopts some idiosyncratic and fussy notation.
indeed a very good book, but I think a better solution for a “dedicated amateur” would be Hinman+Levy+Kunen