Ultraproduct

Summary

An ultraproduct combines a family of structures of the same language into one new structure. An ultrafilter on the index set decides which properties hold in it. A property holds in the ultraproduct when it holds in a "large" set of factors, where "large" means "in the ultrafilter". By Łoś's theorem, this rule works for every first-order formula. Ultraproducts give a short proof of the compactness theorem and build nonstandard models such as the hyperreal numbers.