10
$\begingroup$

I'm currently a first year mathematics graduate student, and am at an institution which does not have any work being done in mathematical logic, or any logicians on the staff. I've taken a mathematical logic course with the textbook by Leary, 'A Friendly Introduction to Mathematical Logic, 2nd. ed.' (I actually took the class with the author at his university), and am comfortable with it's contents.

I am looking for recommendations of graduate mathematical logic textbooks that would reflect the work and content done in a graduate logic course, so that I may see more advanced model theory and proof theory and get a better feel for whether or not these are topics that I would more enjoy studying.

$\endgroup$
5
  • $\begingroup$ The classic Model Theory of Chang and Keisler is a nice one, has undergraduate material but also graduate, and i found it very accesible. A guide to classic and modern model theory is good too. One from Ziegler and Tent, A course in model theory. And a personal favorite: A concise introduction to mathematical logic, by W Rautenberg. $\endgroup$ Commented Oct 18, 2017 at 2:40
  • $\begingroup$ Unfortunately, model theory and proof theory are treated very differently in practice, so that graduate-level books that focus on one of those areas tend to have very little about the other. If you are looking for less-advanced model theory and less-advanced proof theory, it may be possible, but unfortunately there is still not a wide selection of recent texts. $\endgroup$ Commented Oct 18, 2017 at 18:51
  • $\begingroup$ In addition to the excellent list of books give by Peter Smith at logicmatters.net/resources/pdfs/Appendix.pdf , it is also worth looking at Monk's book ( springer.com/us/book/9780387901701 ) which is a graduate-level introduction to the field and has a particularly thorough coverage of complete and incomplete theories. $\endgroup$ Commented Oct 18, 2017 at 18:55
  • $\begingroup$ If that's the case, then some list of books partitioned into those which contain more model theory than proof theory, and those which are flip flopped in that regard, would be wonderful too! :D $\endgroup$ Commented Oct 18, 2017 at 19:36
  • 1
    $\begingroup$ I just hope that this post can serve as a resource guide for people who have experience in mathematical logic and wish to expand their horizons, but don't quite know where to go because they have no logicians locally to ask :) $\endgroup$ Commented Oct 18, 2017 at 19:37

4 Answers 4

Reset to default
11
$\begingroup$

It's hard to find a "one-size-fit-all" book which fits this description, especially since the literature on logic is getting quite advanced. Back in the days, Joseph Shoenfield's Mathematical Logic was (and still is, in some places) a standard reference, and it does contain a fair bit of model theory (up to Ryll-Nardzewski's categoricity theorem, though do note that there are a lot of important results and definitions in the exercises), some proof theory (say, Herbrand's theorem and Gödel's consistency proof, though, sadly, no cut-elimination or Gentzen style consistency proofs), some recursion theory and some set-theory (including one of the first textbook presentations of forcing).

A more up-to-date book in the same style (and quite a mammoth) is Hinman's Fundamentals of Mathematical Logic, which also focuses on almost everything except proof-theory (the model-theoretic part is quite advanced for a general introduction, getting until Morley's theorem).

One option that does include a bit more of proof theory (and a much less of everything else) is Dirk van Dalen's Logic and Structure, which has some very basic model theory (ultraproducts, model completeness) and has a chapter on normalization for natural deduction systems. It's also considerably shorter than the other, so it may be a nice option. In fact, my recommendation would be to skim van Dalen (which, if I remember correctly, is not much more difficult than Leary) to see if you enjoy what you find there. If so, then you can perhaps focus on more specialized books, such as Poizat's A Course in Model Theory or Pohler's Proof Theory.

$\endgroup$
2
$\begingroup$

I haven't looked through the book by Leary that you mentioned, so I don't know how it compares to the following:

  1. Manin
  2. Chiswell & Hodges
  3. Mundici
$\endgroup$
1
$\begingroup$

Disclaimer: I haven't taken a graduate level logic course.

Peter Smith a retired professor, who used to teach logic at the University of Cambridge put up a guide here with a list of books: http://www.logicmatters.net/tyl/

S. C. Kleene's Introduction to Metamathematics got reviewed by Michael Beeson when it got republished and as I recall, Dr. Beeson said that the book still had relevance for graduate students as a starting point. It is also the first book on Peter Smith's list. The most recent review of the book on Amazaon also says: "This 1952 book by Stephen Cole Kleene (1909-1994) is essential for anyone who wants to understand mathematical logic at the graduate level." It looks like it has over 900 citations on CiteSeer (I don't know if that is large for citations in logic), is still getting cited at present.

$\endgroup$
2
  • 1
    $\begingroup$ Kleene's book is a masterpiece, but it would not be an ideal starting point for anyone in logic, in my opinion. Its main use as a continuing reference is for some technical details for proof theory and the T predicate which Kleene worked out in detail. Overall the terminology and the viewpoint in the book are out of step with modern concerns, and I don't know that someone who studied from Kleene's book alone would be ready to talk to people in the field. The same goes for many other classic books, of course. $\endgroup$ Commented Oct 18, 2017 at 18:48
  • 1
    $\begingroup$ I agree with Carl Mummert, and I'd add that Kleene's book probably accumulated a lot of citations because, for about 15 years after its publication, it was the standard reference for mathematical logic (until Shoenfield's book appeared in 1967). $\endgroup$ Commented Oct 18, 2017 at 18:52
1
$\begingroup$

I recommend the following books all of which are very good.

1. (1984) 2021. Ebbinghaus, H.-D., Flum, J. and Thomas, W. Mathematical Logic. 3rd ed. Springer.

2. (2006) 2010. Rautenberg,W. A Concise Introduction to Mathematical Logic, 3rd ed. Springer.

3. 1967. Shoenfield, J. R. Mathematical Logic. Addison-Wesley Publishing Company.

4. (1972) 2001. Enderton, H. A Mathematical Introduction to Logic. 2nd ed. A Harcourt Science and Technology Company.

5. 1976. Monk, J. D. Mathematical Logic. Springer.

6. (1995) 2013. Hodel, R. E. An Introduction to Mathematical Logic. Dover Publications, Inc.

7. 2005. Hinman, P. G. Fundamentals of Mathematical Logic. A K Peters.

8. 2023. Avigad, J. Mathematical Logic and Computation. Cambridge University Press.

9. 2023. Mileti, J. Modern Mathematical Logic. Cambridge University Press.

10. 2024. Open Logic Project. The Open Logic Text. Complete Build. 2024-12-01.

For difficulty of them, I list the ten books as follows: 1 < 2, 3, 4, 5, 6, 8, 9, 10 < 7. Maybe the difficulty of 2, 3, 4, 5, 6, 8, 9, 10 could be distinguished furthermore, but for me they are almost the same. By the way, I've ever investigated almost all of the books on mathematical logic or symbol logic, and give a table to show indicators of many aspects of about 30 good books, while unfortunately I write it into my book in Chinese of which the English version may be published very late.

Furthermore, since you are a graduate in the department of mathematics, the books numbered as 2-7 and 9 are very suitable to you, but 2, 4 and 9 are better as introductory books which are very modern, and the left could be further readings.

$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.