What Is Mathematical Logic?

Mathematical logic has grown from an exotic branch of mathematics into an indispensable tool in computer science as well as other parts of mathematics. This concise book presents the subject of mathematical logic in a lively and approachable fashion although logic can be a formidably abstruse topic, even for mathematicians. This second edition of What is Mathematical Logic?, originally published 50 years ago, deals with important ideas in modern mathematical logic, without the detailed mathematical work required of those with a professional interest in logic. The ideas are set forth simply and clearly in a pleasant style and, despite the book's relative brevity, all the basic material is covered in these pages. Three new chapters have been added, covering automatic theorem proving, logic beyond traditional first order logic, and other logics including intuitionistic, free, and modal logics. Students of computer science and mathematical logic will find it a stimulating introduction and valuable supplement for courses, including current further reading suggestions in this lively area at the intersection of mathematics, philosophy, and computer science.

1 - Historical survey
2 - The completeness of first-order logic
3 - Model theory
4 - Turing machines
5 - Gödel's incompleteness theorems
6 - Computability in practice
7 - Set theory
8 - Beyond first-order logic
9 - Further topics

Guillermo Badia is Senior Lecturer at the University of Queensland, Australia, where he teaches mathematical logic and is on the editorial board of the Springer journal Archive for Mathematical Logic. He has published over 40 papers on formal logic. From 2022 to 2025 his research is supported by a Discovery Early Career Researcher Award of the Australian Research Council. John N. Crossley is Emeritus Professor at Monash University, Australia, where he was Professor of Pure Mathematics and of Logic. He has published extensively on logic, theoretical computer science, history of mathematics, and medieval and Asian history. John C. Stillwell is Emeritus Professor of the University of San Francisco and former Associate Professor at Monash University, Australia. He has been given many awards for his expository writing, including 16 books, on mathematics. He has published extensively on mathematics and logic, particularly from the historical viewpoint.