The book provides a self-contained introduction to mathematical logic and computability theory for students of mathematics or computer science. It is organized around the failures and successes of Hilbert's programme for the formalization of Mathematics. It is widely known that the programme failed with Godel's incompleteness theorems and related negative results about arithmetic. Unfortunately, the positive outcomes of the programme are less well known, even among mathematicians. The book covers key successes, like Godel's proof of the completeness of first-order logic, Gentzen's proof of...