Tame vs. Wild

Abstract

In this chapter we will compare two classical structures: the field of complex numbers $({\mathbb {C}},+,\cdot )$ and the standard model of arithmetic $({\mathbb {N}},+,\cdot )$. The former is vast and mysterious, the latter deceptively simple. As it turns out, as far as the model-theoretic properties of both structures are concerned, the roles are reversed, the former is very tame while the latter quite wild, and those terms have well-understood meanings. In recent years, tameness has become a popular word in model theory. Tameness is not defined formally, but a structure is considered tame if the geometry of its definable sets is well-described and understood. Tameness has different levels. The most tame structures are the minimal ones. All parametrically definable unary relations in a minimal structure are either finite or cofinite. The examples of minimal structures that we have seen so far are the structures with no relations on them—the trivial structures—and $({\mathbb {N}},<)$. It is somewhat surprising that the ultimate number structure—the complex numbers, is also minimal. It is a fascinating example.