Set theory generated by Abelian group theory

Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah’s first independence result regarding the Whitehead problem used established set-theoretical methods (discussed below), his later work required new ideas; it is on these that we focus. We emphasize the nature of the new ideas and the historical context in which they arose, and we do not attempt to give precise technical definitions in all cases, nor to include a comprehensive survey of the algebraic results. In fact, very little algebraic background is needed beyond the definitions of group and group homomorphism. Unless otherwise specified, we will use the word “group” to refer to an abelian group, that is, the group operation is commutative. The group operation will be denoted by +, the identity element by 0, and the inverse of a by−a. We shall use na as an abbreviation for a + a + · · · + a [n times] if n is positive, and na = (−n)(−a) if n is negative. A group is called free if and only if it has a basis, that is, a linearly independent subset which generates the group. (The notions are the same as for vector spaces except that the scalars are from the ring of integers; so not every group is free.) Equivalently, a group is free if and only if it is isomorphic to a direct sumof copies ofZ, the group of integers under addition. A crucial fact is that a subgroup of a free group is free [16, Theorem 14.5]. At a couple of points we will have occasion to refer to groups which are not necessarily commutative. In this context—the variety of all groups—the characterization of “free” is different; in fact, with the exceptions of the trivial group (of cardinality 1) and of Z, free groups in this variety are not commutative. However, it is still the case that subgroups of free groups are free [28, p. 95]. The Whitehead problem asks whether a certain necessary condition for a group to be free (defined in Section 1) is also sufficient. The almost