The Prime Number Conspiracy: The Biggest Ideas in Math from Quanta (The MIT Press) | An Illustrated Theory of Numbers | Elements of Mathematics: From Euclid to GÃ¶del | Mathematics Rebooted: A Fresh Approach to Understanding | The Calculus Story: A Mathematical Adventure | A Most Elegant Equation: Eulerâs Formula and the Beauty of Mathematics | Ten Great Ideas about Chance

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**Product Description**:

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 thousand 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 analysisâfinding the âright axiomsâ to prove fundamental theoremsâand 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. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the âright axiomâ to prove it.

By using a minimum of mathematical logic in a well-motivated way, *Reverse Mathematics* will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

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