Ultraproducts

Ultraproducts are a construction in model theory that produce new structures by combining a family of structures along an ultrafilter. Given a family M_i of structures in the same language L indexed by I, and an ultrafilter U on I, the ultraproduct ∏_i M_i / U is formed by taking the cartesian product ∏_i M_i and modding out by the equivalence relation f ~ g if { i ∈ I : f(i) = g(i) } ∈ U.

Ultrapowers are a special case where all factors M_i are equal to a fixed M; with I = N (or any index set) and a fixed M, one writes M^I / U. If U is principal and concentrates at i0 ∈ I, the ultraproduct is isomorphic to M_{i0}.

Łoś's theorem asserts that every first-order sentence φ in the language L is true in the ultraproduct ∏_i M_i / U exactly when the set { i ∈ I : M_i ⊨ φ } belongs to U. This yields that ultraproducts preserve the common first-order theory: if each M_i ⊨ T, then ∏_i M_i / U ⊨ T.

Ultraproducts have several consequences and applications. They provide a method to construct nonstandard models, especially ultrapowers with nonprincipal ultrafilters, such as the hyperreal numbers *R, which extend the real field by infinitesimals. They are used to transfer properties, study saturation and stability, and give model-theoretic proofs of compactness and other theorems.

Limitations include that only first-order properties are preserved, and different ultrafilters can yield nonisomorphic ultrapowers. The construction is central to model theory and nonstandard analysis.