This paper is a survey of results in the program of reverse mathematics which deal with theorems from algebra. The goal of reverse mathematics is to establish the axiomatic strength of theorems of ordinary mathematics by proving their equivalence to set-existence axioms in the context of subsystems of second-order arithmetic: in algebra this means that only countable algebraic structures are considered.
In the first section the six subsystems of second-order arithmetic that are most relevant from the reverse mathematics viewpoint are introduced, and the proofs of their relationships are sketched. In the second section the authors list many theorems of countable algebra that were found to be equivalent to one of these subsystems over a weaker subsystem. For two of these results the authors give proofs, which rely on (standard) reverse mathematics results. For the other theorems appropriate references to the literature are given. S. G. Simpson’s monograph [Subsystems of second order arithmetic (Perspectives in Mathematical Logic, Springer, Berlin) (1999; Zbl 0909.03048)], which is still listed as “in preparation” in the paper under review, is the basic reference in the area and contains proofs for most of the results listed in this survey.
In the third and last section the authors compare Reverse Algebra and Recursive Algebra: the different goals of the two subjects are explained, and the ways in which results in one area can be used to obtain results in the other area are illustrated.
For the entire collection see [Zbl 0905.03002].