Mathematical Logic and related book suggestions.

Question

I'm majoring in mathematics, and have a deep interest in logic (-related) fields.

I have studied Robert Causey's Logic, Sets And Recursion, so I think I have had a basic knowledge of sentential calculus and predicate calculus.

Now I am very interested in topics on answering the questions such like

• "what axioms truly are?"
• "what does a definition truly mean?"
• "what is the the definition of equality $=$"
• what is the basis of all mathematics?"
• "what is a formal system?"

These question seem to fall in many fields of mathematics, such as mathematical logic, metamathematics, type theory, model theory, category theory. (I'm not sure whether I'm correct.)

So if I want to learn this and get a complete understanding of these, which books would you recommend to me to read next, and in what order? (I'm afraid that I accidentally read the advanced books without having read the basic books yet.)

And I'm curious about the differences between type theory, modal theory and metamathematics; maybe I don't need to study these all to answer my curious questions stated above.

I've looked at brief introductions of these subject in the wikipedia, but still don't understand it very well.

Is it good to read Introduction to Metamathematics by Kleene?

Answer 1

There are indeed many sources, and trying to proceed in the direction from more basic to more advanced makes it all the more difficult.

I've got a couple of recommendations to start with:

1. A Mathematical Introduction to Logic (Herbert B. Enderton)
2. Mathematical Logic, (H,D. Ebbinghaus, J.Flum, W. Thomas)
3. Foundations of Mathematical Logic by Haskell B. Curry
4. Introduction to Metamathematics (by Kleene)

Other great resources are suggested on Wikipedia for each topic you've mentioned, usually at the bottom of the topic's dedicated entry.

See also the website logic matters, a great site maintained by Peter Smith of the University of Cambridge. The linked page gives you access to a pdf study guide, and also access to a pdf list of books/authors that Peter Smith lists, discusses, and recommends (foundations, math logic, etc...)