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Reverse Mathematics - Proofs from the Inside Out (Hardcover)
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Reverse Mathematics - Proofs from the Inside Out (Hardcover)
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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 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
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. 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 "right axiom" 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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