Buy from Amazon

George S Boolos,John P Burgess and Richard C Jeffrey
Computability and logic
Many books on mathematical logic soon plunge the reader into a mass of difficult to understand symbols. Computability and logic by Boolos, Burgess and Jeffrey starts off in an easier to follow style, and so gives the reader a gentler learning curve for the subject. The book starts off with a look at Turing machines, showing how they can be considered equivalent to any other computer. This is followed by an introduction to the theory of recursive functions. The second part of the book deals with first order logic, leading to the proof of Gödel's incompleteness theorem and related results.
The third part of the book moves on to more advanced topics such as second order logic and nonstandard models.
The authors generally don't give full proofs of theorems, but rather give a sketch which will enable the readers to complete the proof for themselves. This not only enables more material to be covered, but is done in a way that I felt enhanced the readers' understanding of the topics, and so the book is useful as a reference to the important considerations in the proofs. The book is aimed at those studying mathematical logic at undergraduate level, but I felt that it would also be useful for those wanting to undertake independent study of the subject.