Ordered Fields

Real analysis begins with the construction and axiomatization of the real number system. Before defining the real numbers, we introduce the algebraic structure of an ordered field, which captures the arithmetic and order properties that $\mathbb{Q}$ and $\mathbb{R}$ share.

Definition and basic properties

Definition1.1Ordered field

An ordered field is a field $F$ equipped with a subset $P \subseteq F$ (called the positive elements) satisfying:

(Trichotomy) For all $x \in F$ , exactly one of the following holds: $x \in P$ , $x = 0$ , or $-x \in P$ .
(Closure under addition) If $x, y \in P$ , then $x + y \in P$ .
(Closure under multiplication) If $x, y \in P$ , then $xy \in P$ .

We write $x > 0$ to mean $x \in P$ , and $x < y$ to mean $y - x \in P$ .

RemarkTotal order

The relation $<$ defined above is a total order on $F$ : it is transitive, antisymmetric, and satisfies trichotomy. Moreover, it is compatible with the field operations: if $a < b$ and $c \in F$ , then $a + c < b + c$ ; if $c > 0$ , then $ac < bc$ .

ExampleThe rationals

The field $\mathbb{Q}$ is an ordered field with $P = \{x \in \mathbb{Q} \mid x > 0\}$ . The positive rationals are closed under addition and multiplication, and trichotomy holds by the usual ordering inherited from the integers.

ExampleC is not an ordered field

The field $\mathbb{C}$ of complex numbers cannot be made into an ordered field. Suppose there were such an order. Then either $i > 0$ or $-i > 0$ (by trichotomy). If $i > 0$ , then $i^2 = -1 > 0$ , contradicting that $-1 < 0$ . Similarly, $-i > 0$ leads to a contradiction. Thus no total order compatible with the field operations exists on $\mathbb{C}$ .

ExampleFinite fields

Every ordered field has characteristic zero. Indeed, if $F$ has characteristic $p > 0$ , then $1 + 1 + \cdots + 1 = 0$ ( $p$ times), contradicting closure of $P$ under addition. In particular, no finite field can be ordered.

Absolute value and distance

Definition1.2Absolute value

For $x$ in an ordered field $F$ , the absolute value of $x$ is

$|x| = \begin{cases} x & \text{if } x \geq 0, \\ -x & \text{if } x < 0. \end{cases}$

Theorem1.1Triangle inequality

For all $x, y$ in an ordered field $F$ :

$|x + y| \leq |x| + |y|.$

RemarkReverse triangle inequality

A useful variant is $||x| - |y|| \leq |x - y|$ , which follows from applying the triangle inequality to $x = (x - y) + y$ and $y = (y - x) + x$ .

ExampleBasic absolute value identities

For all $x, y \in F$ :

$|x| \geq 0$ , with equality iff $x = 0$ .
$|-x| = |x|$ .
$|xy| = |x| \cdot |y|$ .
$|x|^2 = x^2$ .
If $|x| < \epsilon$ , then $-\epsilon < x < \epsilon$ .

These properties make $|x - y|$ a natural candidate for a distance function (metric) on $F$ .

Archimedean and non-Archimedean fields

Definition1.3Archimedean property

An ordered field $F$ is Archimedean if for all $x, y \in F$ with $x > 0$ , there exists $n \in \mathbb{N}$ such that $nx > y$ .

RemarkDensity of rationals in R

The Archimedean property is equivalent to saying: there are no "infinitely large" or "infinitely small" elements. In $\mathbb{R}$ , it implies that $\mathbb{Q}$ is dense in $\mathbb{R}$ : between any two real numbers, there exists a rational.

ExampleQ is Archimedean

Given $x, y \in \mathbb{Q}$ with $x > 0$ , choose $n \in \mathbb{N}$ such that $n > y/x$ (such $n$ exists by the well-ordering of $\mathbb{N}$ ). Then $nx > y$ . Thus $\mathbb{Q}$ is Archimedean.

ExampleNon-Archimedean fields

The field $\mathbb{Q}(t)$ of rational functions over $\mathbb{Q}$ , ordered by declaring $t$ to be "infinitely large" (i.e., $t > n$ for all $n \in \mathbb{N}$ ), is a non-Archimedean ordered field. In this field, no multiple of $1$ exceeds $t$ .

Similarly, the field of surreal numbers contains infinitesimals (elements $\epsilon > 0$ with $\epsilon < 1/n$ for all $n \in \mathbb{N}$ ) and is non-Archimedean.

Theorem1.2Characterization of Archimedean fields

An ordered field $F$ is Archimedean if and only if $\mathbb{Q}$ is dense in $F$ : for all $x < y$ in $F$ , there exists $q \in \mathbb{Q}$ with $x < q < y$ .

Supremum and completeness

Not every ordered field has the completeness property that characterizes the real numbers.

Definition1.4Supremum (least upper bound)

Let $S \subseteq F$ be a nonempty subset of an ordered field $F$ . An element $b \in F$ is an upper bound of $S$ if $x \leq b$ for all $x \in S$ .

If $S$ has an upper bound, we say $S$ is bounded above. The supremum of $S$ , denoted $\sup S$ , is the least upper bound: an element $\beta \in F$ such that:

$\beta$ is an upper bound of $S$ , and
if $\alpha$ is any upper bound of $S$ , then $\beta \leq \alpha$ .

If $\sup S$ exists, it is unique.

Definition1.5Complete ordered field

An ordered field $F$ is complete (or has the least upper bound property) if every nonempty subset $S \subseteq F$ that is bounded above has a supremum in $F$ .

ExampleQ is not complete

Let $S = \{x \in \mathbb{Q} \mid x^2 < 2\}$ . Then $S$ is nonempty and bounded above in $\mathbb{Q}$ (e.g., $2$ is an upper bound), but $S$ has no supremum in $\mathbb{Q}$ . The "least upper bound" would have to satisfy $\alpha^2 = 2$ , but $\sqrt{2} \notin \mathbb{Q}$ . Thus $\mathbb{Q}$ is incomplete.

RemarkCharacterization of the reals

The real numbers $\mathbb{R}$ are the unique complete ordered field (up to isomorphism). The completeness axiom — every nonempty bounded-above set has a supremum — is what distinguishes $\mathbb{R}$ from $\mathbb{Q}$ and is the foundation for limits, continuity, and the entire theory of real analysis.

Summary

Ordered fields provide the algebraic and order-theoretic framework for real analysis:

$\mathbb{Q}$ is an Archimedean ordered field, but not complete.
$\mathbb{R}$ is the complete Archimedean ordered field.
Completeness ensures that Cauchy sequences converge, continuous functions on $[a,b]$ attain their maximum, and bounded monotone sequences converge.

The next concepts will develop the construction of $\mathbb{R}$ via Dedekind cuts or Cauchy sequences, and establish the completeness axiom rigorously.