Review by Qi Feng
Translation by Min Chen and Connie Yu

Review of Subsystems of Second Order Arithmetic

Stephen G. Simpson
Pennsylvania State University
Subsystems of Second Order Arithmetic
1999, 445pp.
ISBN 3-540-64882-8

This book is a monograph of the series Perspectives in Mathematical Logic, published by Springer, the result of the author's work of nearly fifteen years. The book provides a systematic and comprehensive report of the work of the school of Reverse Mathematics, led by Simpson and H. Friedman, in foundations of mathematics, especially on the partial realization of ``Hilbert's Program.'' Many previously unpublished research results, detailed references, and open problems can be found in the book.

This book deals with many problems in foundations of mathematics of the following type: which logical systems correspond to which mathematical theorems? In other words, which set existence axioms are necessary to prove many theorems of classical mathematics? This book provides us with a hierarchy of subsystems of second order arithmetic which is useful to understand the logical structure of core mathematics. What is second order arithmetic? First order arithmetic or Peano arithmetic deals only with natural numbers. This system has besides the well known mathematical axioms also the induction scheme. Second order arithmetic has, in addition to first order arithmetic, variables for subsets of the natural numbers. The term ``order'' refers to the level of the variables.

This book separates second order arithmetic into a hierarchy of subsystems of increasing logical strength. It starts with the most elementary system RCA_0. Some theorems of ordinary mathematics of fundamental importance are placed at the appropriate level of the hierarchy. On the one hand, a theorem is proved in a certain subsystem; on the other hand, axioms of the subsystem are provable from that theorem reversely. This method provides a kind of measurement of logical complexity of these fundamental mathematical theorems.

The book consists of four parts and ten chapters. The first part consists of only Chapter One. This chapter is the introduction of the book and defines the system of second order arithmetic. The second part, from Chapter Two to Chapter Six, forms the main body of the book. In this part, we see the treatment of topics of ordinary mathematics such as complete separable metric space, Banach space, countable structures of algebra, vector space, ordinary differential equations, fixed points, infinite games, combinatorics, and many other fields. The third part, from Chapter Seven to Chapter Nine, is the core of the book. In this part, tools of mathematical logic are used to reveal the logical nature of the hierarchy of many subsystems. It explains why these subsystems are natural, reasonable, and necessary, as well as the independence of the subsystems. The fourth part is the chapter on additional results. Here, interested readers can find many unsolved problems at the frontier of research.

Qi Feng, Professor, Institute of Mathematics, Chinese Academy of Sciences