UltraPowers

Ultrapowers are a fundamental construction in model theory used to build new structures from a given structure by combining it with an ultrafilter. Let M be a structure in a language L, I an index set, and U a nonempty ultrafilter on I. The ultrapower M^I/U is the quotient of the direct product M^I by the equivalence relation x ~ y if {i in I : x(i) = y(i)} ∈ U. The constant sequences c_m(i) = m define an embedding j: M → M^I/U by m ↦ [c_m]. This embedding is elementary, so M ≺ M^I/U, and M^I/U is elementarily equivalent to M.

Los’s theorem provides the key transfer principle: for any first-order L-sentence φ, M^I/U ⊨ φ if and only if {i ∈ I : M ⊨ φ} ∈ U. Consequently, ultrapowers preserve all first-order truths of M, while typically enlarging the universe to include new, “nonstandard” elements.

Nonprincipal ultrafilters (those not concentrating on a single index) yield ultrapowers that are usually proper elementary extensions, containing elements not present in M. A classic example is the ultrapower of the natural numbers, which gives a nonstandard model of arithmetic with infinite and infinitesimal elements.

Cardinality considerations depend on M and I, but ultrapowers are at least as large as the index set and often much larger. In model theory, ultrapowers are closely related to ultraproducts and are used to study saturation properties, Keisler’s order, and various model-theoretic phenomena. They provide a flexible, canonical way to transfer structures into richer models while preserving first-order theories.