o-minimal structures on the field of real numbers. (English) Zbl 0854.03036

This is a survey of recent work on \(o\)-minimality. Many results are discussed, but none proved. No background in model theory is needed to read the paper. The notion of \(o\)-minimality is defined only for the reals, and an \(o\)-minimal structure on the field of real numbers is presented as a family of sets of \(n\)-tuples (for all \(n)\) with additional properties. There is a discussion of consequences of \(o\)-minimality (for example the Monotonicity, Cell Decomposition, and Triangulation Theorems), a description of some classes of \(o\)-minimal structures and Miller’s exponential growth rate dichotomy theorem, and some comments on applications.


03C52 Properties of classes of models
12L12 Model theory of fields
12D99 Real and complex fields
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations