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.
