Question on Mathematical Logic & Book Recommendation

Question

I'm in the last year of my undergraduate studies in mathematics and I would appreciate your oppinion in the following thought.

Should I study Mathematical Logic, if I want to do further studies in Algebra and/or applied Algebra? Will this be useful?

And if the answer is yes, could you please recommend me a good book for Mathematical Logic, which containing the following?

• Propositional Calculus: The Language of Propositional Calculus. Truth Values, appreciation, logical implications. Axiomatization of propositional calculus, completeness. Independence of axioms.

• Predicate Calculus: 1st order languages. Structures, Models, Truth. Axiomatization of 1st order Predicate Calculus, completeness.

And at last, do you have further advises for how to study this course?

Thank you in advance.

Answer 1

For the topics you've listed, Logic and Structure by van Dalen is a good, math-oriented book that you should be able to work through by yourself.

Doing some logic will help you better understand what proofs are, which is good if you're a mathematician! But you should not really expect it to help you directly solve research problems (in algebra, or other branches of pure math) except for very specific occasional problems that turn out to have slick logic-based proofs, that logicians will be happy to brag about if you ask them to. :) However, the "logic" that's used is inevitably harder than what you'll get from a first pass at propositional and predicate logic. For that, you should work through something like van Dalen first, then if your research interests require it, learn more model theory/proof theory/descriptive set theory/whatever at some later point.

Answer 2

If you are interested at learning mathematical logic, I strongly recommend that you check out Richard Hodel's An Introduction to Mathematical Logic. It covers everything you mention (some in the form of lengthy exercises with hints), the exposition is admirably clear, and, most importantly, it's cheap, since it has been re-published by Dover! I've found it immensely useful and it's one of my favorite logic books.

As for whether mathematical logic will help you with algebra, I'd say that there is a nice interplay here between algebraic and logical ideas. On the one hand, algebraic ideas in many cases serve either as inspiration or proto-types for properly logical ideas (for instance, I was always struck by the analogies between the construction of the splitting field for a polynomial and Henkin's term models), and, on the other hand, getting the hang of logic may help you to get a deeper understanding of the underlying algebra (here, I found Robinson's ideas of model completeness, especially as developed by Manders, to be very enlightening). Of course, this rarely will lead you to a new result, but the resulting clarity may be useful.