section_path: Formal definitions/Axiomatic approach
section_name: Axiomatic approach
paragraph_text: Axiomatic approach
  
  Let R {\displaystyle \mathbb {R} } denote the set of all real numbers. Then:
  
  The last property applies to the real numbers but not to the rational numbers (or to other more exotic ordered fields). For example, { x ∈ Q: x 2 < 2 } {\displaystyle \{x\in \mathbb {Q}:x^{2}<2\}} has a rational upper bound (e.g., 1.42), but no least rational upper bound, because 2 {\displaystyle {\sqrt {2}}} is not rational.
  
  These properties imply the Archimedean property (which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound N; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N.
  
  The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R 1 {\displaystyle \mathbb {R} _{1}} and R 2 {\displaystyle \mathbb {R} _{2}}, there exists a unique field isomorphism from R 1 {\displaystyle \mathbb {R} _{1}} to R 2 {\displaystyle \mathbb {R_{2}} }. This uniqueness allows us to think of them as essentially the same mathematical object.
  
  For another axiomatization of R {\displaystyle \mathbb {R} }, see Tarski's axiomatization of the reals.