Promotional Material for Simpson's Book


Subsystems of Second Order Arithmetic


Stephen G. Simpson
Department of Mathematics
Pennsylvania State University

Publication Data

Disciplines Addressed

Subject Matter

Simpson's book is an original contribution to foundations of mathematics, with emphasis on the role of set existence axioms. Part A demonstrates that many familiar theorems of algebra, analysis, functional analysis, and combinatorics are logically equivalent to the axioms needed to prove them. This phenomenon is known as Reverse Mathematics. Subsystems of second order arithmetic based on such axioms correspond to several well known foundational programs: finitistic reductionism (Hilbert), constructivism (Bishop), predicativism (Weyl), and predicative reductionism (Feferman/Friedman). Part B is a thorough study of models of these and other systems. The book includes an extensive bibliography and a detailed index.

Author's Area of Expertise

Simpson has published extensively in mathematical logic and foundations of mathematics. He is a recognized authority on subsystems of second order arithmetic and their role in foundations of mathematics.

An Endorsement

Here is a statement from Harvey Friedman, to be used in advertising and promoting the book.

From: Harvey Friedman <>
Date: Tue, 30 Jun 1998 09:42:21 +0100

From 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.

Competing Literature

Simpson's book is unique. Most of the material in it has not previously appeared in book form. Much of it has not previously been published in any form. There is no other book on second order arithmetic, or on subsystems of second order arithmetic, or on reverse mathematics.

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.

Keywords for Catalog Index

Mathematics Subject Classification Numbers

Promotional Activities

Web Page

The book's web page is It contains the front matter and chapter one in DVI, PDF, and PostScript formats. Eventually it will have a list of open problems, links to new research papers, etc.

Mailing List

From September 1997 onward, Simpson has been moderator of an automated e-mail discussion group on foundations of mathematics, at The name of the group is FOM and there are currently more than 300 subscribers, all professionals in this area. Simpson will use FOM to foment interest in this book and related philosophical/mathematical issues.


Journals for Review

Professional Societies   /   7 Dec 1998