Introduction to mathematical logic

My main reason for recommending Elliot Mendelson's Introduction to mathematical logic to someone wishing to study mathematical logic is the start of the third chapter (of 5). This chapter is on formal number theory, and starts off with a list of axioms and then some proofs. That's what mathematical logic is all about isn't it? Well maybe it's more about metaproofs - that is proofs about what you can prove, and certainly that is the main content of this book. However, I feel that having a few pages of the proofs you are dealing with is vital to give the student a foothold in this difficult subject, but, somewhat surprisingly, its difficult to find such proofs in books at this level.

The first chapter is on the (relatively simple) propositional calculus. The second deals with quantification theory - I would recommend that you look at the start of chapter 3 to start with, so as to have a concrete example. The later parts of chapter 3 get on to more complicated mathematical logic, such as Gödel's incompleteness theorem. Chapter 4 is on set theory. Now you may have come across set theory at school, but I have to tell you that the axiomatic version is a totally different ball game. But with the introduction to axiomatics from the previous chapters, this book would be a good place to start studying it. The fifth chapter is on computability. I wouldn't suggest this book as a starting point for this - it's much easier to approach it via real computers. However, it could be useful for seeing the links with other forms of logic.