Archimedean property


In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
Roughly speaking, it is the property of having no infinitely large or infinitely small elements.
It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.
The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.
An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean.
A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean.
For example, a linearly ordered group that is Archimedean is an Archimedean group.
This can be made precise in various contexts with slightly different formulations.
For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.

History and origin of the name of the Archimedean property

The concept was named by Otto Stolz after the ancient Greek geometer and physicist Archimedes of Syracuse.
The Archimedean property appears in Book V of Euclid's Elements as Definition 4:
Because Archimedes credited it to Eudoxus of Cnidus it is also known as the "Theorem of Eudoxus" or the Eudoxus axiom.
Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.

Definition for linearly ordered groups

Let and be positive elements of a linearly ordered group G.
Then is infinitesimal with respect to if, for every natural number, the multiple is less than, that is, the following inequality holds:
The group is Archimedean if there is no pair such that is infinitesimal with respect to.
Additionally, if is an algebraic structure with a unit — for example, a ring — a similar definition applies to.
If is infinitesimal with respect to 1, then is an infinitesimal element.
Likewise, if is infinite with respect to 1, then is an infinite element.
The algebraic structure is Archimedean if it has no infinite elements and no infinitesimal elements.

Ordered fields

s have some additional properties:
In this setting, an ordered field is Archimedean precisely when the following statement, called the axiom of Archimedes, holds:
Alternatively one can use the following characterization:

Definition for normed fields

The qualifier "Archimedean" is also formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows.
Let be a field endowed with an absolute value function, i.e., a function which associates the real number 0 with the field element 0 and associates a positive real number with each non-zero and satisfies
and.
Then, is said to be Archimedean if for any non-zero there exists a natural number such that
Similarly, a normed space is Archimedean if a sum of terms, each equal to a non-zero vector, has norm greater than one for sufficiently large.
A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality,
respectively.
A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean.
The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.

Examples and non-examples

Archimedean property of the real numbers

The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function when, the more usual, and the -adic absolute value functions.
By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value.
The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete.
The completion with respect to the usual absolute value is the field of real numbers.
By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field.
On the other hand, the completions with respect to the other non-trivial absolute values give the fields of -adic numbers, where is a prime integer number ; since the -adic absolute values satisfy the ultrametric property, then the -adic number fields are non-Archimedean as normed fields.
In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows.
Denote by the set consisting of all positive infinitesimals.
This set is bounded above by 1.
Now assume for a contradiction that is nonempty.
Then it has a least upper bound, which is also positive, so.
Since is an upper bound of and is strictly larger than, is not a positive infinitesimal.
That is, there is some natural number for which.
On the other hand, is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal between and, and if then is not infinitesimal.
But, so is not infinitesimal, and this is a contradiction.
This means that is empty after all: there are no positive, infinitesimal real numbers.
The Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.

Non-Archimedean ordered field

For an example of an ordered field that is not Archimedean, take the field of rational functions with real coefficients.

To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations.
Now if and only if f − g > 0, so we only have to say which rational functions are considered positive.
Call the function positive if the leading coefficient of the numerator is positive.
By this definition, the rational function 1/x is positive but less than the rational function 1.
In fact, if is any natural number, then n = n/x is positive but still less than 1, no matter how big is.
Therefore, 1/x is an infinitesimal in this field.
This example generalizes to other coefficients.
Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field.
Taking the coefficients to be the rational functions in a different variable, say, produces an example with a different order type.

Non-Archimedean valued fields

The field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values.
All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.

Equivalent definitions of Archimedean ordered field

Every linearly ordered field contains the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of, which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid.
The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in.
The following are equivalent characterizations of Archimedean fields in terms of these substructures.
1. The natural numbers are cofinal in. That is, every element of is less than some natural number. Thus an Archimedean field is one whose natural numbers grow without bound.
2. Zero is the infimum in of the set.
3. The set of elements of between the positive and negative rationals is non-open. This is because the set consists of all the infinitesimals, which is just the set when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. Observe that in both cases, the set of infinitesimals is closed. In the latter case, every infinitesimal is less than every positive rational, there is neither a greatest infinitesimal nor a least positive rational, and there is nothing else in between. Consequently, any non-Archimedean ordered field is both incomplete and disconnected.
4. For any in the set of integers greater than has a least element.
5. Every nonempty open interval of contains a rational.
6. The rationals are dense in with respect to both sup and inf. Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.