"How did Gödel establish the two Theorems of Incompleteness, and why do they matter? Smith (U. of Cambridge) advises readers to take their time in answering these and related questions he poses as he presents a variety of proofs for the First Theorem and shows how to prove the Second. He also examines a group of related results with the same care and attention to detail. In 36 well-paced chapters Smith builds his case from a basic introduction to G<:o>del's theorems on to such issues as the truths of arithmetic, formalized arithmetics, primitive recursive functions, identifying the diagonalization Lemma in the First Theorem and using it, dirivability conditions in the Second Theorem. Turing machines (and recursiveness) and the Church-Turing thesis. Accessible without being dismissive, this is accessible to philosophy students and equally suitable for mathematics students taking a first course in logic."
"... Without doubt, a mandatory reference for every philosopher interested in philosophy of mathematics. The text is, in general, written in a prose style but without avoiding formalisms. It is very accurate in the mathematical arguments and it offers to mathematicians and logicians a detailed approach to Gödel's theorems, covering many aspects which are not easy to find in other presentations."
Reinhard Kahle, Mathematical Reviews
What are Gödel's Theorems, how were they established and why do they matter? Written with great clarity, this book is accessible to philosophy students with a limited formal background. It is equally valuable to mathematics students taking a first course in mathematical logic.