Starting with his results on the Whitehead Problem in 1973, Shelah’s work has led to some powerful set-theoretic methods for analyzing the structure of modules, even, in many cases, without going beyond ZFC. For example, just this year they have led to the proof in ZFC that every Baer module is projective.

We will survey the developments from a contemporary perspective, in terms of properties of cotorsion pairs, a homologically defined notion which will be explained. The key method is to analyze membership in a cotorsion pair in terms of small modules.

Paul Eklof