> The reals are the unique Cauchy-complete Archimedian ordered field; every other set with those properties is isomorphic to the reals. One can also arrive at

> The reals are the unique Cauchy-complete Archimedian ordered field; every other set with those properties is isomorphic to the reals. One can also arrive at the reals by constructing from the rationals through completion. https://en.wikipedia.org/wiki/Complete_metric_space#Completion https://en.wikipedia.org/wiki/Complete_metric_space#Completi... Edit: There is a nice wiki page about the different approaches https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...