Tame vs. Wild

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 are concerned, the roles are reversed, the former is tame while the latter quite wild, and those terms have precise 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. In many respects it is one of the most intriguing mathematical structures. The real field is not minimal, and this is due to the fact that its ordering is definable, but it is order-minimal and all order-minimal structures are considered tame. No other number structures that include addition and multiplication discussed in this book are tame. They are all on the wild side and we will see glimpses of this wilderness in the second section of this short chapter.