example, the compactness theorem can be proved this way. In the special case of ultrapowers, one gets elementary extensions of structures. For example, in nonstandard analysis, the hyperreal numbers can be constructed as an ultraproduct of the real numbers, extending the domain of discourse from real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead the functions and relations are defined "pointwise modulo" , where is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If is nonprincipal, then the extension thereby obtained is nontrivial. In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This construction yields a rigorous way to consider , that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces. Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter. In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for aggregating the preferences of infinitely many individuals. Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (for example, Kirman and Sondermann, 1972). Mihara (1997, 1999) shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable. See also Notes References Bibliography Further reading Order theory Families of sets Nonstandard analysis