Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers

The foundation of mathematics is not found in a single discipline since it is a general way of thinking in a very rigorous logical fashion.  This book was written especially for readers who are about to make their first contact with this very way of thinking.
Chapters 1–5 provide a rigorous, self contained construction of the familiar number systems (natural numbers, integers, real, and complex numbers) from the axioms of set theory. This construction trains readers in many of the proof techniques that are ultimately used almost subconsciously. In addition to important applications, the author discusses the scientific method in general (which is the reason why civilization has advanced to today’s highly technological state), the fundamental building blocks of digital processors (which make computers work), and public key encryption (which makes internet commerce secure).  The book also includes examples and exercises on the mathematics typically learned in elementary and high school. Aside from serving education majors, this further connection of abstract content to familiar ideas explains why these ideas work so well.
Chapter 6 provides a condensed introduction to abstract algebra, and it fits very naturally with the idea that number systems were expanded over and over to allow for the solution of certain types of equations.
Finally, Chapter 7 puts the finishing touches on the excursion into set theory.  The axioms presented there do not directly impact the elementary construction of the number systems, but once they are needed in an advanced class, readers will certainly appreciate them.
Chapter coverage includes: Logic; Set Theory; Number Systems I: Natural Numbers; Number Systems II: Integers; Number Systems III: Fields; Unsolvability of the Quintic by Radicals; and More Axioms.