Almost all of the problems studied in this book are motivated by an
overriding foundational question: What are the appropriate axioms
for mathematics? Through a series of case studies, these axioms are
examined to prove particular theorems in core mathematical areas
such as algebra, analysis, and topology, focusing on the language
of second-order arithmetic, the weakest language rich enough to
express and develop the bulk of mathematics. In many cases, if a
mathematical theorem is proved from appropriately weak set
existence axioms, then the axioms will be logically equivalent to
the theorem. Furthermore, only a few specific set existence axioms
arise repeatedly in this context, which in turn correspond to
classical foundational programs. This is the theme of reverse
mathematics, which dominates the first half of the book. The second
part focuses on models of these and other subsystems of
second-order arithmetic.
General
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