Model theory
Minimality

In every structure, every finite subset { a 1, …, a n } {\displaystyle \{a_{1},\dots,a_{n}\}} is definable with parameters: Simply use the formula

Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is also always definable.

This leads to the concept of a minimal structure. A structure M {\displaystyle {\mathcal {M}}} is called minimal if every subset A ⊆ M {\displaystyle A\subseteq {\mathcal {M}}} definable with parameters from M {\displaystyle {\mathcal {M}}} is either finite or cofinite.The corresponding concept at the level of theories is called strong minimality:A theory T is called strongly minimal if every model of T is minimal.A structure is called strongly minimal if the theory of that structure is strongly minimal. Equivalently, a structure is strongly minimal if every elementary extension is minimal.Since the theory of algebraically closed fields has quantifier elimination, every definable subset of an algebraically closed field is definable by a quantifier-free formula in one variable. Quantifier-free formulas in one variable express Boolean combinations of polynomial equations in one variable, and since a nontrivial polynomial equation in one variable has only a finite number of solutions, the theory of algebraically closed fields is strongly minimal.

On the other hand, the field R {\displaystyle \mathbb {R} } of real numbers is not minimal: Consider, for instance, the definable set

This defines the subset of non-negative real numbers, which is neither finite nor cofinite.One can in fact use φ {\displaystyle \varphi } to define arbitrary intervals on the real number line.It turns out that these suffice to represent every definable subset of R {\displaystyle \mathbb {R} }. This generalisation of minimality has been very useful in the model theory of ordered structures.A densely totally ordered structure M {\displaystyle {\mathcal {M}}} in a signature including a symbol for the order relation is called o-minimal if every subset A ⊆ M {\displaystyle A\subseteq {\mathcal {M}}} definable with parameters from M {\displaystyle {\mathcal {M}}} is a finite union of points and intervals.
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