Ultraproduct - Ultralimit

For the ultraproduct of a sequence of metric spaces, see Ultralimit.

In model theory and set theory, an ultralimit or limiting ultrapower is a direct limit of a sequence of ultrapowers.

Beginning with a structure, A₀, and an ultrafilter, D₀, form an ultrapower, A₁. Then repeat the process to form A₂, and so forth. For each n there is a canonical diagonal embedding . At limit stages, such as A_ω, form the direct limit of earlier stages. One may continue into the transfinite.