Ebook: The Foundations of Mathematics

I: THE INTUITIVE BACKGROUND ; 1. Mathematical Thinking ; 2. Number Systems ; THE BEGINNINGS OF FORMALISATION ; 3. Sets ; 4. Relations ; 5. Functions ; III THE DEVELOPMENT OF AXIOMATIC SYSTEMS ; 8. Natural Numbers and Proof by Induction ; 9. Real Numbers ; 10. Real Numbers as a Complete Ordered Field ; 1 Complex Numbers and Beyond ; IV: USING AXIOMATIC SYSTEMS ; 12. Axiomatic Structures and Structure Theorems ; 13. Permutations and Groups ; 14. Infinite Cardinal Numbers ; 15. Infinitesimals ; V: STRENGTHENING THE FOUNDATIONS ; 16. Axioms for Set Theory;The transition from school to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. This book bridges the divide.