What are some recommended books on mathematical logic?

[brainy kevin]

Sorry to have two threads up at the top of the Science Book Discussion forum, but I couldn't find a thread for this. I'm interested in learning some mathematical logic. Here are the books I'm considering, please tell me what you think of them or suggest better alternatives.

Mathematical Logic - Kleene. What I plan to get. I would consider getting Tarski's book as well. Seems to be a good introduction to logic, and complete (Logic pun, get it?) and a very reasonable price.

Introduction to Metamathematics - Beeson. I don't know much about it, but it seems like it's good. Any suggestions for a god meta-math book?

First order logic - Smullyan. Supposed to be good but hard.

Axiomatic Set Theory - Suppes. I understand naive set theory pretty well, and this looks like a good course in Axiomatic Set theory.

Set Theory and Its Philosophy - Looks pricey, but supposed to be good. Might consider it.

Axiomatic Set Theory - Bernays. Supposed to be a well thought out look at set theory.

Principia Mathematica - Russell and Whitehead. Maybe someday! Well, at least publishers have stopped sniffing glue and are selling the three volumes for $50 rather than the previous price of $700(!) for the set. Or maybe the copyright expired. (Seriously, $700!)

So there you have it! Please tell me if my books are good, or recommend alternatives. I would also like some recommendations on other types of logic, like modal logic and whatever. Some books on model theory would be nice. What else is there to logic?

 

[xristy]

If the request is for mathematical logic as opposed to set theory, then for an introduction that includes treatment of both the proof theory and the model theory of first order logic consider https://www.amazon.com/dp/0387942580/?tag=pfamazon01-20. Set Theory and Its Philosophy is interesting but not an introduction.

 

[Reedeegi]

Two books on categorical logic, a sort of meta-logical theory derived from category theory, that are very insightful are Toposes, Triples, and Theories and Topoi: The Categorial Analysis of Logic. I prefer the Categorical Approach to logic as it takes less statements as axioms than logic or other mathematical theories.