 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 ordered or normed groups, fields, and other algebraic structures. 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 'The sphere and the cylinder'^{[1]}.
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 nonzero 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 nonzero elements, one of which is infinitesimal with respect to the other, is said to be nonArchimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group.
This can be made precise in various contexts with slightly different ways of formulation. 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.
Contents
History and origin of the name of the Archimedean property
The concept is named after the ancient Greek geometer and physicist Archimedes of Syracuse.
The Archimedean property appears in Book V of Euclid's Elements as Definition 4:
Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.Because Archimedes credited it to Eudoxus of Cnidus it is also known as the "Theorem of Eudoxus"^{[2]} 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 x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y (or equivalently, y is infinite with respect to x) if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds:
The group G is Archimedean if there is no pair x,y such that x is infinitesimal with respect to y.
Additionally, if K is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to K. If x is infinitesimal with respect to 1, then x is an infinitesimal element. Likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements.
Ordered fields
An ordered field has some additional nice properties.
 One may assume that the rational numbers are contained in the field.
 If x is infinitesimal, then 1/x is infinite, and vice versa. Therefore to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements.
 If x is infinitesimal and r is a rational number, then rx is also infinitesimal. As a result, given a general element c, the three numbers c/2, c, and 2c are either all infinitesimal or all noninfinitesimal.
In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds:
 Let x be any element of K. Then there exists a natural number n such that n > x.
Alternatively one can use the following characterization:
 For any positive ε in K, there exists a natural number n, such that 1/n < ε.
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 F 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  x  with each nonzero and satisfies  xy  =  x   y  and . Then, F is said to be Archimedean if for any nonzero there exists a natural number n such that
Similarly, a normed space is Archimedean if a sum of n terms, each equal to a nonzero vector x, has norm greater than one for sufficiently large n. 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, or , respectively. A field or normed space satisfying the ultrametric triangle inequality is called nonArchimedean.
The concept of a nonArchimedean normed linear space was introduced by A. F. Monna.^{[3]}
Examples and nonexamples
Archimedean property of the real numbers
The field of the real numbers is Archimedean both as an ordered field and as a normed field. It is chiefly because the real numbers are obtained as the completion of the rational numbers, which themselves satisfy the axiom in both senses, with respect to an absolute value structure compatible with the ordering.
In the axiomatic theory of real numbers, the nonexistence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows. Denote by Z the set consisting of all positive infinitesimals. This set is bounded above by 1. Now assume by contradiction that Z is nonempty. Then it has a least upper bound c, which is also positive, so c/2 < c < 2c. Since c is an upper bound of Z and 2c is strictly larger than c, 2c must be strictly larger than every positive infinitesimal. In particular, 2c cannot itself be an infinitesimal, for then 2c would have to be greater than itself. Moreover since c is the least upper bound of Z, c/2 must be infinitesimal. But 2c and c/2 cannot have different types by the above result, so there is a contradiction. The conclusion follows that Z is empty after all: there are no positive, infinitesimal real numbers.
One should note that the Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.
NonArchimedean ordered field
For an example of an ordered field that is not Archimedean, take the field of rational functions with real coefficients. (A rational function is any function that can be expressed as one polynomial divided by another polynomial; we will assume in what follows that this has been done in such a way that the leading coefficient of the denominator is positive.) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now f > g 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. (One must check that this ordering is well defined and compatible with addition and multiplication.) By this definition, the rational function 1/x is positive but less than the rational function 1. In fact, if n is any natural number, then n(1/x) = n/x is positive but still less than 1, no matter how big n 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 nonArchimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say y, produces an example with a different order type.
NonArchimedean valued fields
The field of the rational numbers endowed with the padic metric and the padic 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.^{[4]} There is a nontrivial nonArchimedean valuation on every infinite field.
Equivalent definitions of Archimedean ordered field
Every linearly ordered field K contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of K, 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 K. The following are equivalent characterizations of Archimedean fields in terms of these substructures.^{[5]}
1. The natural numbers are cofinal in K. That is, every element of K is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound.
2. Zero is the infimum in K of the set {1/2, 1/3, 1/4, …}. (If K contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.)
3. The set of elements of K between the positive and negative rationals is closed. This is because the set consists of all the infinitesimals, which is just the closed set {0} when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between, a situation that points up both the incompleteness and disconnectedness of any nonArchimedean field.
4. For any x in K the set of integers greater than x has a least element. (If x were a negative infinite quantity every integer would be greater than it.)
5. Every nonempty open interval of K contains a rational. (If x is a positive infinitesimal, the open interval (x,2x) contains infinitely many infinitesimals but not a single rational.)
6. The rationals are dense in K with respect to both sup and inf. (That is, every element of K is the sup of some set of rationals, and the inf of some other set of rationals.) 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.
Notes
 ^ G. Fisher (1994) in P. Ehrlich(ed.), Real Numbers, Generalizations of the Reals, and Theories of continua, 107145, Kluwer Academic
 ^ Knopp, Konrad (1951). Theory and Application of Infinite Series (English 2nd ed.). London and Glasgow: Blackie & Son, Ltd.. p. 7. ISBN 0486661652.
 ^ Monna, A. F., Over een lineare Padisches ruimte, Indag. Math., 46 (1943), 74–84.
 ^ Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990. ISBN 082478412X
 ^ Schechter 1997, §10.3
References
 Schechter, Eric (1997). Handbook of Analysis and its Foundations. Academic Press. ISBN 0126227608. http://www.math.vanderbilt.edu/~schectex/ccc/.
Categories: Field theory
 Ordered groups
 Real algebraic geometry

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