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This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thoUSnd years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis - finding the "right axioms" to prove fundamental theorems - and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation

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"Filling an important niche, this book gives readers a good picture of the basics of reverse mathematics while suggesting several directions for further reading and study. It provides a context for the questions investigated by reverse mathematics and makes an extended argument for their significance within contemporary mathematical practice."--Denis Hirschfeldt, University of Chicago

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John Stillwell is professor of mathematics at the University of San Francisco and an affiliate of the School of Mathematical Sciences at Monash University, Australia. His many books include Mathematics and Its History and Elements of Mathematics: From Euclid to Gödel (Princeton).

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