Mathematical logic has been in existence as a recognised branch of mathematics for over a hundred years. Its methods and theorems have shown their applicability not just to philosophical studies in the foundations of mathematics (perhaps their original raison d'être) but also to ‘mainstream mathematics’ itself, such as the infinitesimal analysis of Abraham Robinson, or the more recent applications of model theory to algebra and algebraic geometry.

Nevertheless, these logical techniques are still regarded as somewhat ‘difficult’ to teach, and possibly rather unrewarding to the serious mathematician. In part, this is because of the notation and terminology that still survives as a relic of the original reason for the subject, and also because of the off-putting and didactically unnecessary logical precision insisted on by some of the authors of the standard undergraduate textbooks. This is coupled by the professional mathematician's very reasonable distrust of so much emphasis on ‘inessential’ non-mathematical details when he or she only requires an insight into the mathematics behind it and straightforward statements of the main mathematical results.

This book presents the material usually treated in a first course in logic, but in a way that should appeal to a suspicious mathematician wanting to see some genuine mathematical applications. It is written at a level suitable for an undergraduate, but with additional optional sections at the end of each chapter that contain further material for more advanced or adventurous readers. The core material in this book assumes as prerequisites only: basic knowledge of pure mathematics such as undergraduate algebra and real analysis; an interest in mathematics; and a willingness to discover and learn new mathematical material.