Construction using hyperreal numbers
Construction using hyperreal numbers [ edit ] As in the hyperreal numbers , one constructs the hyperrationals * Q from the rational numbers by means of an ultrafilter . [10] [11] Here a hyperrational is by definition a ratio of two hyperintegers . Consider the ring B of all limited (i.e. finite) elements in * Q . Then B has a unique maximal ideal I , the infinitesimal numbers. The quotient ring B/I gives the field R of real numbers [ citation needed ] . Note that B is not an internal set in * Q . Note that this construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by the axiom of choice . It turns out that the maximal ideal respects the order on * Q . Hence the resulting field is an ordered field. Completeness can be proved in a similar way to the con...