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The Foundations of Mathematics


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Table of Contents

I: The Intuitive Background 1: Mathematical Thinking 2: Number Systems II: 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 11: 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

About the Author

Ian Stewart is Emeritus Professor of Mathematics at the University of Warwick. He remains an active research mathematician and is a Fellow of the Royal Society. Famed for his popular science writing and broadcasting, for which he is the recipient of numerous awards, his bestselling books include: Does God Play Dice?, Nature's Numbers, and Professor Stewart's Cabinet of Mathematical Curiosities. He also co-authored The Science of Discworld series with Terry Pratchett and Jack Cohen David Tall is Emeritus Professor of Mathematical Thinking at the University of Warwick. Internationally known for his contributions to mathematics education, his most recent book is How Humans Learn to Think Mathematically (2013).


`give[s] a very comprehensive overview of the mathematics involved in part of a first year mathematics degree. It would be of most value to tutors and lecturers in encouraging them to think about teaching methodologies and how they might modify them to help students in their thinking of mathematical ideas and their development of a coherent structure of mathematical concepts.' John Sykes, Maths in School `Review from previous edition There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book. ' The Bulletin of Mathematics Books

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