The Foundations of Mathematics

Stewart and Tall have written some excellent mathematical texts, and this book is one of them. It deals with the basics of mathematics and treats them in a clear but thorough way. Although it is probably not essential, it is a good accompaniment to anybody starting a mathematics degree. They take the approach that one should already be familiar with an idea (via examples) before seeing its definition. This works well, although sometimes the effect is that the pace is slow and the exposition is drawn out.

This book was written for those inquisitive/advanced high school students (6th formers in England) to give them an introduction to the number systems, formalization in mathematics and the concept of proof. Stewart is well-known for his clear writing and this book is an excellent example.