Type (model theory)

In model theory and related areas of mathematics, a type is an object that, loosely speaking, describes how a (real or possible) element or elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x₁, x₂,…, x_n that are true of a sequence of elements of an L-structure ${\displaystyle {\mathcal {M}}}$. Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure ${\displaystyle {\mathcal {M}}}$. The question of which types represent actual elements of ${\displaystyle {\mathcal {M}}}$ leads to the ideas of saturated models and omitting types.

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