Department of Mathematics

Pennsylvania State University

http://www.personal.psu.edu/t20/

- Springer-Verlag
- November 1998, XIV + 445 pages
- ISBN 3-540-64882-8
- list price: 98 German marks, or 60 US dollars
- discount: 30 percent for ASL members (promotion code S206)

- Foundations of Mathematics
- Mathematical Logic
- Philosophy of Mathematics
- Mathematics

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:

- Hilbert/Bernays, Grundlagen der Mathematik, Springer, 1939, 1970
- Kleene, Introduction to Metamathematics, Van Nostrand, 1952

Books on first order arithmetic are:

- Hajek/Pudlak, Metamathematics of First-Order Arithmetic, Springer, 1993.
- Kaye, Models of Peano Arithmetic, Oxford, 1991.

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.

- foundations of mathematics
- second order arithmetic
- mathematical logic
- constructivism
- predicativism

- 03F35: Second- and higher-order arithmetic and fragments
- 03B30: Foundations and axiomatics of classical theories

- Association for Symbolic Logic (http://www.aslonline.org/)
- LICS and related conferences (http://www.bell-labs.com/topic/conferences/lics/)
- Kurt Gödel Society (http://www.logic.tuwien.ac.at/kgs/home.html)

- Journal of Symbolic Logic
- Bulletin of Symbolic Logic
- Notre Dame Journal of Formal Logic
- Bulletin of the American Mathematical Society
- Advances in Mathematics
- Philosophia Mathematica
- History and Philosophy of Logic

- Association for Symbolic Logic
- Kurt Gödel Society
- American Mathematical Society
- American Philosophical Association

t20@psu.edu / 7 Dec 1998