The real numbers are a fundamental structure in the study of mathematics. The real numbers are a mathematical set with the properties of a complete ordered field. While these properties identify a number of facts, not all of them are essential to completely define the real numbers. The real numbers can either be defined axiomatically as a complete ordered field, or can be reduced by set theory as a set of all limits of Cauchy sequences of rational numbers (a completion of a metric space). Either way, the constructions produce field-isomorphic sets.