Mathematical logic | Model theory | Topology | Real algebraic geometry

O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure (M,<,...) which is totally ordered by < is called an o-minimal structure if and only if every definable subset X ⊂ M (with parameters taken from M) is a finite union of intervals and points. O-minimality can be regarded as a weak form of quantifier elimination. A structure M is o-minimal if and only if every formula with one free variable and parameters in M is equivalent to a quantifier-free formula involving only the ordering, also with parameters in M. This is analogous to the minimal structures, which are exactly the analogous property down to equality. A theory T is an o-minimal theory if every model of T is o-minimal. It is known that the complete theory T of an o-minimal structure is an o-minimal theory. This result is remarkable because, in contrast, the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure which is not minimal. (Wikipedia).

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Real closed field | Quantifier elimination | Hassler Whitney | Wilkie's theorem | Total order | Pfaffian function | Model theory | Theory (mathematical logic) | Weakly o-minimal structure | Exponential function | Stratification (mathematics) | C-minimal theory | Strongly minimal theory | Tame topology | Łojasiewicz inequality | Union (set theory) | Real number | Semialgebraic set | Complete theory | Structure (mathematical logic) | Mathematical logic | Analytic function | Interval (mathematics) | Real algebraic geometry | Definable set | Boolean algebra (structure)