Pseudofinite

Pseudofinite is a term in model theory used to describe infinite structures that are, in a precise sense, indistinguishable from finite ones by first-order logic. A structure M is pseudofinite if it is infinite and elementarily equivalent to an ultraproduct of finite structures. In practice, many examples come from taking an ultraproduct of finite structures, such as finite fields, with respect to a non-principal ultrafilter.

The most studied instances are pseudofinite fields. A field K is pseudofinite if it arises as an ultraproduct of finite fields. Pseudofinite fields are sometimes called quasi-finite fields. They share several finite-field-like properties in a first-order sense, yet they are infinite objects. A classical characterization due to Ax says that a field is pseudofinite precisely when it is perfect, pseudo algebraically closed (PAC), and has exactly one extension of K of each finite degree n.

Properties and theories: The theory of a pseudofinite structure is the common first-order theory obtained as the ultraproduct limit of the theories of the finite structures used in the construction. This makes pseudofinite structures natural laboratories for transferring intuition from finite structures to infinite ones. However, being pseudofinite does not imply that the structure itself is finite; the structure is infinite but satisfies many first-order properties that hold in finite structures.

Related concepts and context: Ultraproducts and Łoś’s theorem are foundational to the construction and analysis of pseudofinite structures. The study of pseudofinite fields has yielded complete axiomatizations of their theory and clarified how finite-field phenomena extend to the infinite, via model-theoretic methods. See also ultraproduct, model theory, finite model theory, quasi-finite fields.