 Strongly minimal theory

A complete theory with infinite models is strongly minimal when, in every model, each definable subset in one variable, with parameters allowed, is finite or cofinite. The requirement applies to all models and all parameter choices. The infinite-universe condition excludes trivial finite theories.

For an infinite set in the language of equality, a one-variable formula with finitely many parameters can only distinguish those named elements from the remaining elements, so every definable subset is finite or cofinite. In an algebraically closed field, quantifier elimination likewise makes every one-variable definable subset finite or cofinite, since a nonzero polynomial has finitely many roots. A dense linear order without endpoints fails this property: a proper open interval and its complement are both infinite. Strong minimality is a particularly restrictive source of omega-stability.

In model theory, a branch of mathematical logic, a theory is termed "strongly minimal" if it satisfies certain specific properties related to definable sets.