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 this is a book about the foundations of mathematics—a topic once of interest to outstandingmathematicians, such asDedekind, Poincaré, and Hilbert, but today sadly neglected. this neglect is unfortunate for several reasons: • As mathematics splits into more and more specialties, the need for a unifying viewpoint becomes more acute. • Foundations unify not only mathematics but also the neighboring disciplines of computer science and physics. • Recent advances in mathematical logic throw new light on the foundations of analysis, and on the elusive concept of mathematical “depth.”  this book aims at the last point in particular, by focusing on the topic of reverse mathematics. As its name suggests, reverse mathematics looks at the concept of proof in the opposite to normal direction. Instead of seeking the consequences of given axioms, it seeks the axioms needed to prove given theorems. this is actually an old idea, at least in the foundations of geometry. Fromthe time of Euclid until the nineteenth century it was a burning question whether the parallel axiom was needed to prove theorems such as the Pythagorean theorem. We review the history of the parallel axiom in chapter 1 of this book, as a case study in reverse mathematical ideas, together with the similar story of the axiom of choice in set theory. Although both these axioms illustrate the idea of reverse mathematics, the subject as it is understood today liesmostly in a narrowbut important region between geometry and set theory: the theory of real numbers, which is the foundation of calculus, analysis, and most of mathematical physics. (Reversemathematics has alsomade interesting contributions to algebra, combinatorics, and topology which we mention more brie_y.)  the real numbers, as we understand them today, emerged fromnineteenth century e_orts to arithmetize analysis and geometry. By building real numbers fromsets of rational numbers (and hence, ultimately, from sets of natural numbers) it becomes possible to encode sequences of real numbers and arbitrary continuous functions—and hencemost of the objects of analysis—by sets of natural numbers.We review the arithmetization of analysis, and also the basic theorems of analysis, in chapters 2 and 3. A er this we are ready to ask: which axioms do we need to prove these basic theorems?  the answer, roughly, is a set of axioms for the natural numbers (the Peano axioms) plus a suitable set existence axiom. Now set existence axioms come in various strengths, depending on the strength of the theorems we wish to prove. the lowest useful strength turns out to be intimately related to the foundations of computation: it asserts the existence of computable sets. this in turn involves a study of the concept of computation, which merges with analysis because both have a common basis in arithmetic. A er an informal introduction to computability in chapter 4 we develop a formal concept of computation, and its arithmetization, in chapter 5. In chapters 6 and 7 we bring together the ideas of analysis, arithmetic, and computation in some axiom systems for analysis, known as RCA0, WKL0, and ACA0. these systems, which are distinguished mainly by set existence axioms of increasing strength, between them provemost of the basic theorems of analysis.More remarkably, they sort the basic theorems into three levels because, once above the “base” level of RCA0, most theorems are equivalent to the set existence axiom of the system that proves them. this makes each of these set existence axioms the “right axiom” in the sense articulated by Friedman (1975): When a theorem is proved from the right axioms, the axioms can be proved from the theorem. We will see, for example, that RCA0 can prove the intermediate value theorem; the de_ning axiom of WKL0 is the right axiom to prove the Heine-Borel theorem and the extreme value theorem; and the de_ning axiom of ACA0 is the right axiom to prove the Cauchy convergence criterion and the Bolzano-Weierstrass theorem.  thus in reverse mathematics we meet the usual cast of characters from an introductory real analysis course, but in an entirely new story.