Complete Type

A complete type (also called a complete n-type) is a maximal consistent set of first-order formulas with parameters in a given structure or over a parameter set, in a fixed tuple of free variables; it assigns, for every formula in those variables, either the formula or its negation, thereby describing all first-order properties that a tuple can satisfy relative to the parameter set.

A complete type over a parameter set A is a maximal consistent set of first-order formulas with parameters from A: for every formula φ(x,a) with parameters a from A, the type contains either φ(x,a) or its negation, and the whole set remains consistent with the theory T.