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I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might be reasonable to cover in a semester. Is it feasible to end with a little model theory, or would that be a pipe dream?

EDIT The students are theoretically math majors, and the title of the course is "Senior Problem Solving", so they are seniors with high probability. Which means that they have a couple of courses in real analysis, and one in abstract algebra.

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Depends on where you start and what student level is. Could you add some more background? There are lots of possibilities, I have taught logic courses at the sophomore level that included a bit of model theory. No fancy stuff since the course was really about proof theory, but they knew what a model was, what they were used for, and they had a firm grasp of soundness and completeness. – François G. Dorais Dec 24 '11 at 21:35
What is the assumed background of students in a topics course? From a search of your department's website there doesn't appear to be a regularly taught introduction to logic course. So this would be a first course in a logic rather than a true topics-in-logic course that follows a first logic course, right? – KConrad Dec 24 '11 at 21:37
@Igor: Of course. Just do not pick any of the classic books like Mendelson or Enderton. These are not suitable any longer. In particular, these are "pre-computer age" books. There are some other suggestions generated by my question, I did not look at them since these came after the semester ended. Probably the best thing to do is to use different text for different topics. Did you look at Manin's book? One of the topics I suggest are Ehrenfeucht games. – Mark Sapir Dec 25 '11 at 16:03
Since Mark mentioned Ehrenfeucht games, I think you may want to take a look at "Models and games", a recent book by Jouko Väänänen. Logic and model theory are introduced and developed through games, and though the book is introductory, some advanced topics are reached by the end.… – Andrés Caicedo Dec 26 '11 at 2:28

2 Answers 2

Kenneth Kunen recently wrote a wonderful introduction to Mathematical Logic, called "The Foundations of Mathematics" (ISBN: 978-1-904987-14-7), published in 2009. The book's only prerequisite is the mathematical maturity that an Introduction to Analysis course would provide, so it sounds like your students would be prepared. The book provides a brief introduction to axiomatic set theory, model theory, and computability theory; and it culminates with a proof of Godel's incompleteness theorems and Tarski's theorem on the non-definability of truth. There are also a couple brief discussions of the philosophy of mathematics; these are given from the perspective of the working mathematician, and they are used to motivate the material. And they are very helpful. In fact, the most salient thing about this book is that it is exceptionally clear, well-written, and easy to learn from. (Kunen also wrote "Set Theory: An Introduction To Independence Proofs" which is also exceptionally clear, well-written, and easy to learn from). Your students will be grateful for the fact that this book is available (new) on for less than $25.

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Thanks, that sounds very good, I will check it out. Manin sounds a little scary for American undergrads... – Igor Rivin Dec 26 '11 at 11:24
@Igor: You are right about Manin and undergrads. Perhaps you can use it only as a source of possible topics to cover. In general there are so many different things you can do in this course that you need to be careful. This course is no "Linear Algebra" in many respects. But you have got some good suggestions here. I wish I had these answers when I asked my question. – Mark Sapir Dec 26 '11 at 23:37

I think it is very doable. I took a class like that as an undergrad. The textbook was Enderton's "Introduction to Mathematical Logic". It did enough Hilbert-style proof theory to get up to the incompleteness theorem, then discussed models, interpretations, Tarski's definition of truth, etc. It seemed great at the time and wasn't a terribly hard course. My gripe these many years later is that I wish it had said something about sequent calculus since that would have changed how I thought of logic if I'd known about it at the time.

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Enderton is too formal and hopelessly outdated, unfortunately. – Mark Sapir Dec 25 '11 at 5:03
@Mark: what do you mean by "outdated"? – Igor Rivin Dec 25 '11 at 15:52
Just compare those books with Manin's book. – Mark Sapir Dec 25 '11 at 18:12

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