0.1 Completeness
The [[rational-numbers|rational numbers]] \(\mathbb{Q}\) and real numbers \(\mathbb{R}\) have similar algebraic and [[order|order]] properties (they are both densely ordered fields). The crucial property that distinguishes \(\mathbb{R}\) from \(\mathbb{Q}\) is its completeness.
There are two main ways to define the completeness of \(\mathbb{R}\), one based on the [[order|order]] properties of \(\mathbb{R}\) and the existence of suprema. The other is based on the metric properties of \(\mathbb{R}\) and the convergence of Cauchy sequences.
0.2 Completeness from Order and Suprema
The following axiomatic property of the real numbers is called Dedekind Completeness. Dedekind (1872) showed that the real numbers are characterized by the condition that they are a complete ordered field.
Every nonempty set of real numbers that is bounded from above has a supremum.
Example: Define \(A\subset \mathbb{Q}\) by \[ A:= \{ x \in \mathbb{Q}: x^{2}<2 \}. \] Then \(A\) is bounded from above by every \(M\in \mathbb{Q}^+\) such that \(M^2>2\). Nevertheless, \(A\) has no supremum in \(\mathbb{Q}\) because \(\sqrt{2 }\) is [[rational-numbers|irrational]].