Real numbers (as a complete ordered field)

The real numbers $\mathbb{R}$ are characterized (up to a unique order-preserving field isomorphism) by the following structure:

• $(\mathbb{R},+,\cdot)$ is a field: $(\mathbb{R},+)$ is an abelian group with identity $0$, $(\mathbb{R}\setminus\{0\},\cdot)$ is an abelian group with identity $1$, and multiplication distributes over addition.

• $\le$ is a total order on $\mathbb{R}$ compatible with the field operations, meaning:

  • if $a\le b$ then $a+c\le b+c$ for all $c\in\mathbb{R}$,
  • if $0\le a$ and $0\le b$ then $0\le ab$.

• (Completeness / least upper bound property) Every nonempty subset $S\subseteq\mathbb{R}$ that is bounded above has a supremum in $\mathbb{R}$.

Completeness is what distinguishes $\mathbb{R}$ from $\mathbb{Q}$ and underlies many limit and convergence theorems in analysis.

Examples:

• The set $S=(0,1)$ is nonempty and bounded above, and $\sup S=1\in\mathbb{R}$.
• The set $S=\{x\in\mathbb{R}: x^2<2\}$ is bounded above and has $\sup S=\sqrt{2}\in\mathbb{R}$.
• In contrast, viewed as a subset of $\mathbb{Q}$, $\{q\in\mathbb{Q}: q^2<2\}$ has no supremum in $\mathbb{Q}$ (illustrating incompleteness of $\mathbb{Q}$).