http://www.personal.psu.edu/t20/
From: Harvey Friedman <friedman@math.ohio-state.edu>
Date: Tue, 30 Jun 1998 09:42:21 +0100From the point of view of the foundations of mathematics, this definitive work by Simpson is the most anxiously awaited monograph for over a decade. The "subsystems of second order arithmetic" provide the basic formal systems normally used in our current understanding of the logical structure of classical mathematics. Simpson provides an encyclopedic treatment of these systems with an emphasis on *Hilbert's program* (where infinitary mathematics is to be secured or reinterpreted by finitary mathematics), and the emerging *reverse mathematics* (where axioms necessary for proving theorems are determined by deriving axioms from theorems). The classical mathematical topics treated in these axiomatic terms are very diverse, and include standard topics in complete separable metric spaces and Banach spaces, countable groups, rings, fields, and vector spaces, ordinary differential equations, fixed points, infinite games, Ramsey theory, and many others. The material, with its many open problems and detailed references to the literature, is particularly valuable for proof theorists and recursion theorists. The book is both suitable for the beginning graduate student in mathematical logic, and encyclopedic for the expert.
Other books on foundations of mathematics are:
Books on first order arithmetic are:
Unlike the books of Hilbert/Bernays and Kleene, Simpson's book is much more focused on classical mathematical practice. In addition, this book delves much more deeply than others into subsystems of second order arithmetic and the model theory of such systems.
t20@psu.edu / 7 Dec 1998