nLab Archimedean ordered field

Idea

An Archimedean ordered field is an ordered field that satisfies the archimedean property.

Definition

Using field homomorphisms from the rationals

The rational numbers are the initial ordered field, so for every ordered field $F$ there is a field homomorphism $h:\mathbb{Q}\to F$. Since every field homomorphism between ordered fields is an injection, the rational numbers $\mathbb{Q}$ is a subset of the ordered field $F$, and we can suppress the field homomorphism via coercion, such that given $q \in \mathbb{Q}$ one can derive that $q \in F$. Thus, $F$ is Archimedean if for all elements $x \in F$ and $y \in F$, if $x \lt y$, then there exists a rational number $q\in \mathbb{Q}$ such that $x \lt q$ and $q \lt y$.

Using Dedekind cut-like conditions

A real number in an ordered field $F$ is an element $x \in F$ which satisfies the Dedekind cut axioms:

1. there exists a rational number $q \in \mathbb{Q}$ such that $q \lt x$
2. there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r$
3. for all rational numbers $q \in \mathbb{Q}$ and $q' \in \mathbb{Q}$, $q \lt q'$ and $q' \lt x$ implies that $q \lt x$
4. for all rational numbers $r \in \mathbb{Q}$ and $r' \in \mathbb{Q}$, $x \lt r'$ and $r' \lt r$ implies that $x \lt r$
5. for all rational numbers $q \in \mathbb{Q}$, $q \lt x$ implies that there exists a rational number $q' \in \mathbb{Q}$, such that $q \lt q'$ and $q' \lt x$
6. for all rational numbers $r \in \mathbb{Q}$, $x \lt r$ implies that there exists a rational number $r' \in \mathbb{Q}$, such that $x \lt r'$ and $r' \lt r$
7. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt x$ and $x \lt r$ implies that $q \lt r$
8. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt r$ implies that $q \lt x$ or $x \lt r$

We have the following results:

• The first and second conditions say that every element $x \in F$ is bounded below and above by rational numbers, and thus strictly not an infinite element. This also implies that there are no infinitesimal elements, because there are no element $x \in F$ whose multiplicative inverse is an infinite element.

• The fifth and sixth conditions independently imply that every element $x \in F$ is strictly not an infinitesimal element.

These four conditions together imply the archimedean property for the ordered field $F$.

• The third, fourth, and seventh conditions are always true for all elements $x \in F$ because of transitivity of the strict order relation.

• Finally, the eighth condition says that every element $x \in F$ is located, and is true for all elements $x \in F$ because the pushout of the open intervals $(q, \infty)$ and $(-\infty, r)$ with canonical inclusions $(q, r) \to (q, \infty)$ and $(q, r) \to (-\infty, r)$ is equivalent to $F$ itself.

Thus, an Archimedean ordered field is an ordered field $F$ where every element $x \in F$ is a real number.

Note that this definition is not the same as saying that $F$ contains every real number - the latter definition results in the Dedekind real numbers, which is the union of all Archimedean ordered fields and the terminal Archimedean ordered field.

Properties

Every Archimedean ordered field is a dense linear order. This means that the Dedekind completion of every Archimedean ordered field is the field of all real numbers.

Dedekind cuts

Every element $x \in F$ in an Archimedean ordered field satisfies the axioms of Dedekind cuts:

1. there exists a rational number $q \in \mathbb{Q}$ such that $q \lt x$
2. there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r$
3. for all rational numbers $q \in \mathbb{Q}$ and $q' \in \mathbb{Q}$, $q \lt q'$ and $q' \lt x$ implies that $q \lt x$
4. for all rational numbers $r \in \mathbb{Q}$ and $r' \in \mathbb{Q}$, $x \lt r'$ and $r' \lt r$ implies that $x \lt r$
5. for all rational numbers $q \in \mathbb{Q}$, $q \lt x$ implies that there exists a rational number $q' \in \mathbb{Q}$, such that $q \lt q'$ and $q' \lt x$
6. for all rational numbers $r \in \mathbb{Q}$, $x \lt r$ implies that there exists a rational number $r' \in \mathbb{Q}$, such that $x \lt r'$ and $r' \lt r$
7. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt x$ and $x \lt r$ implies that $q \lt r$
8. for all rational numbers $q \in \mathbb{Q}$ and $r \in \mathbb{Q}$, $q \lt r$ implies that $q \lt x$ or $x \lt r$

We have the following results:

• The first condition is always true because for all $x \in F$, we have $x - 1 \in F$, and by the Archimedean principle there exists a rational number $q \in \mathbb{Q}$ such that $x - 1 \lt q \lt x$.

• The second condition is always true because for all $x \in F$, we have $x + 1 \in F$, and by the Archimedean principle there exists a rational number $r \in \mathbb{Q}$ such that $x \lt r \lt x + 1$.

• The fifth condition is always true because for all $x \in F$ and $q \in \mathbb{Q}$, if $q \lt x$, then by the Archimedean principle there exists a rational number $q' \in \mathbb{Q}$ such that $q \lt q' \lt x$.

• The sixth condition is always true because for all $x \in F$ and $r \in \mathbb{Q}$, if $x \lt r$, then by the Archimedean principle there exists a rational number $q' \in \mathbb{Q}$ such that $x \lt r' \lt r$.

• The third, fourth, and seventh conditions are always true for all elements $x \in F$ because of transitivity of the strict order relation.

• Finally, the eighth condition says that every element $x \in F$ is located, and is true for all elements $x \in F$ because the union $(q, \infty) \cup (-\infty, r)$ of the open intervals $(q, \infty)$ and $(-\infty, r)$ is the improper subset of $F$.

Continuous and differentiable structure

Every Archimedean ordered field is a differentiable space:

Pointwise continuous functions

Let $F$ be an Archimedean ordered field. A function $f:F \to F$ is continuous at a point $c \in F$ if

$$isContinuousAt(f, c) \coloneqq \forall \epsilon \in (0, \infty). \forall x \in F. \exists \delta \in (0, \infty). (\vert x - c \vert \lt \delta) \implies (\vert f(x) - f(c) \vert \lt \epsilon)$$

$f$ is pointwise continuous in $F$ if it is continuous at all points $c$:

$$isPointwiseContinuous(f) \coloneqq \forall c \in F. isContinuousAt(f, c)$$

The set of all pointwise continuous functions is defined as

$$C^0(F) \coloneqq \{f \in F \to F \vert isPointwiseContinuous(f)\}$$

Pointwise differentiable functions

Let $F$ be an Archimedean ordered field. A function $f:F \to F$ is differentiable at a point $c \in F$ if

$$isDifferentiableAt(f, c) \coloneqq isContinuousAt(f, c) \times \exists L \in F. \forall \epsilon \in (0, \infty). \forall x \in F. \exists \delta \in (0, \infty). \forall h \in (-\delta, 0) \cup (0, \delta). \left| \frac{f(c + h) - f(c)}{h} - L \right| \lt \epsilon$$

$f$ is pointwise differentiable in $F$ if it is differentiable at all points $c$:

$$isPointwiseDifferentiable(f) \coloneqq \forall c \in F. isDifferentiableAt(f, c)$$

The set of all pointwise differentiable functions is defined as

$$D^0(F) \coloneqq \{f \in F \to F \vert isPointwiseDifferentiable(f)\}$$

Admissibility

Let $\Sigma \subseteq \Omega$ be a sub-$sigma$-frame of the frame of truth values. Then an Archimedean ordered field $F$ is admissible for $\Sigma$ if the pseudo-order $(-)\lt(-):F \times F \to \Omega$ restricts to a binary function $(-)\lt(-):F \times F \to \Sigma$.

Category of Archimedean ordered fields

The category of Archimedean ordered fields is the category whose objects are Archimedean ordered fields and whose morphisms are ring homomorphisms between Archimedean ordered fields. (Every ring homomorphism between Archimedean ordered field can be proven to be a strictly monotonic field homomorphism).

The category of Archimedean ordered fields is a thin category. It is also a skeletal category and a gaunt category, and impredicatively is the subset of the power set of real numbers which consists of all the Archimedean ordered subfields of the real numbers.

The initial object in the category of Archimedean ordered fields is the rational numbers and the terminal object in the category of Archimedean ordered fields is the (Dedekind) real numbers.

More generally, one can consider, for every sub-$\sigma$-frame $\Sigma \subseteq \Omega$ of the frame of truth values, the subcategory of Archimedean ordered fields which are admissible for $\Sigma$ and ring homomorphisms between Archimedean ordered fields admissible for $\Sigma$. While the rational numbers are still the initial object in this category, the terminal object in this object is a version $\mathbb{R}_\Sigma$ of the Dedekind real numbers, which are constructed using Dedekind cuts valued in $\Sigma$. In the larger category of all Archimedean ordered fields, the ring homomorphism $\mathbb{R}_\Sigma \hookrightarrow \mathbb{R}_D$ is not provably an isomorphism.

Examples

Archimedean ordered fields include

• rational numbers
• real closed fields
• real numbers

In constructive mathematics, one has the different notions of real numbers

• Cauchy real numbers (terminal Archimedean ordered field where every element merely has a locator)
• HoTT book real numbers (initial Cauchy complete Archimedean ordered field)
• Dedekind real numbers (terminal Archimedean ordered field, also initial Dedekind complete Archimedean ordered field)

These notions of real numbers are the same if every Dedekind real number merely has a locator, so that the Cauchy real numbers are Dedekind complete. Both excluded middle and countable choice imply that every Dedekind real number has a locator.

Non-Archimedean ordered fields include

• finite fields

• complex numbers

• p-adic numbers

• surreal numbers

Related concepts

• archimedean valued field

• archimedean integral domain

• archimedean group

• Archimedean ordered Artinian local ring

• Archimedean ordered reduced local ring

• Archimedean ordered local integral domain

• differentiable space

• Archimedean ordered field setoid

• admissible Archimedean ordered field

References

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

The definition of the Archimedean property for an ordered field is given in section 4.3 of

• Auke B. Booij, Analysis in univalent type theory (2020) $[$etheses:10411, pdf$]$

The real numbers are defined as the terminal Archimedean ordered field and the complete Archimedean ordered field in:

• Fred Richman, Real numbers and other completions, Mathematical Logic Quarterly 54 1 (2008) 98-108 [doi:10.1002/malq.200710024]