What is a Structure Theory?

Readers of the Abstracts of the American Mathematical Society who wandered into the 'Mathematical logic and foundations' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consisted of equalities and inequalities between uncountable cardinals. In my experience most mathematicians find uncountable cardinals depressing, if they have any reaction to them at all. In fact Shelah was quite right to be so happy, but not because of his cardinal inequalities. He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic. That is what I want to discuss here. In brief, Shelah's work is about the notion of a class of structures which has a good structure theory. We all have a rough intuitive notion of what counts as a good structure theory. For example the structure theory of finitely generated abelian groups is surely a good one. It takes the form of a structure theorem. According to this theorem, each such group G is determined by a set of invariants, namely, the number of cyclic summands of this or that form in a direct sum representation of G. On the other hand, nobody has ever offered a decent structure theory for abelian groups in general; they are just too complicated and too various. Shelah calls his field of study classification theory. The name is a kind of pun, because there are two different levels of classification involved. On the lower level, a structure theory for a class K is about classifying the structures in K. But on a higher level we can classify classes K according to whether or not they have structure theories. Shelah's main theorem (sometimes called his dichotomy theorem) is on this second level. He looks at mathematically defined classes K, and he shows that any such class must either (a) have a structure theory of a certain form, which he explains, or (b) be too complicated to have any decent structure theory. This programme and its completion are wholly and solely the work of Shelah himself. But for ten years or so, an army of model theorists have been at work on the edges of the programme. Above all I should mention Alistair Lachlan at Simon Fraser and Boris Zil'ber in Kemerovo, who have looked in greater depth at narrower ranges of classes with good structure theories. Lachlan and his group have tested and sharpened their intuitions by proving a number of new structure theorems for classes of graphs. (For example, Lachlan [1982, 1984a] and his survey [198-], Woodrow [1979], Lachlan & Woodrow [1980]—the diagrams in this last paper are an artistic masterpiece. Henson [1971] is an earlier contribution in the same area.)