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. The property, as typically construed, states that given two positive numbers x {\displaystyle x} and y {\displaystyle y} , there is an integer n {\displaystyle n} such that n x > y {\displaystyle nx>y} . It also means that the set of natural numbers is not bounded above. 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.
The concept was named by Otto Stolz (in the 1880s) 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.
Let x and y be positive elements of a linearly ordered group G. Then x {\displaystyle x} is infinitesimal with respect to y {\displaystyle y} (or equivalently, y {\displaystyle y} is infinite with respect to x {\displaystyle x} ) if, for any natural number n {\displaystyle n} , the multiple n x {\displaystyle nx} is less than y {\displaystyle y} , that is, the following inequality holds: x + ⋯ + x ⏟ n terms < y . {\displaystyle \underbrace {x+\cdots +x} _{n{\text{ terms}}}<y.\,}
This definition can be extended to the entire group by taking absolute values.
The group G {\displaystyle G} is Archimedean if there is no pair ( x , y ) {\displaystyle (x,y)} such that x {\displaystyle x} is infinitesimal with respect to y {\displaystyle y} .
Additionally, if K {\displaystyle K} is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to K {\displaystyle K} . If x {\displaystyle x} is infinitesimal with respect to 1 {\displaystyle 1} , then x {\displaystyle x} is an infinitesimal element. Likewise, if y {\displaystyle y} is infinite with respect to 1 {\displaystyle 1} , then y {\displaystyle y} is an infinite element. The algebraic structure K {\displaystyle K} is Archimedean if it has no infinite elements and no infinitesimal elements.
Ordered fields have some additional properties:
In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds:
Alternatively one can use the following characterization: ∀ ε ∈ K ( ε > 0 ⟹ ∃ n ∈ N : 1 / n < ε ) . {\displaystyle \forall \,\varepsilon \in K{\big (}\varepsilon >0\implies \exists \ n\in N:1/n<\varepsilon {\big )}.}
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 K {\displaystyle K} be a field endowed with an absolute value function, i.e., a function which associates the real number 0 {\displaystyle 0} with the field element 0 and associates a positive real number | x | {\displaystyle |x|} with each non-zero x ∈ K {\displaystyle x\in K} and satisfies | x y | = | x | | y | {\displaystyle |xy|=|x||y|} and | x + y | ≤ | x | + | y | {\displaystyle |x+y|\leq |x|+|y|} . Then, K {\displaystyle K} is said to be Archimedean if for any non-zero x ∈ K {\displaystyle x\in K} there exists a natural number n {\displaystyle n} such that | x + ⋯ + x ⏟ n terms | > 1. {\displaystyle |\underbrace {x+\cdots +x} _{n{\text{ terms}}}|>1.}
Similarly, a normed space is Archimedean if a sum of n {\displaystyle n} terms, each equal to a non-zero vector x {\displaystyle x} , has norm greater than one for sufficiently large n {\displaystyle 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, | x + y | ≤ max ( | x | , | y | ) , {\displaystyle |x+y|\leq \max(|x|,|y|),} 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.
The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function | x | = 1 {\displaystyle |x|=1} , when x ≠ 0 {\displaystyle x\neq 0} , the more usual | x | = x 2 {\textstyle |x|={\sqrt {x^{2}}}} , and the p {\displaystyle p} -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 p {\displaystyle p} -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 (from the order) 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 p-adic numbers, where p {\displaystyle p} is a prime integer number (see below); since the p {\displaystyle p} -adic absolute values satisfy the ultrametric property, then the p {\displaystyle p} -adic number fields are non-Archimedean as normed fields (they cannot be made into ordered 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 Z {\displaystyle Z} the set consisting of all positive infinitesimals. This set is bounded above by 1 {\displaystyle 1} . Now assume for a contradiction that Z {\displaystyle Z} is nonempty. Then it has a least upper bound c {\displaystyle c} , which is also positive, so c / 2 < c < 2 c {\displaystyle c/2<c<2c} . Since c is an upper bound of Z {\displaystyle Z} and 2 c {\displaystyle 2c} is strictly larger than c {\displaystyle c} , 2 c {\displaystyle 2c} is not a positive infinitesimal. That is, there is some natural number n {\displaystyle n} for which 1 / n < 2 c {\displaystyle 1/n<2c} . On the other hand, c / 2 {\displaystyle c/2} is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal x {\displaystyle x} between c / 2 {\displaystyle c/2} and c {\displaystyle c} , and if 1 / k < c / 2 ≤ x {\displaystyle 1/k<c/2\leq x} then x {\displaystyle x} is not infinitesimal. But 1 / ( 4 n ) < c / 2 {\displaystyle 1/(4n)<c/2} , so c / 2 {\displaystyle c/2} is not infinitesimal, and this is a contradiction. This means that Z {\displaystyle Z} 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.
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 {\displaystyle f>g} if and only if f − g > 0 {\displaystyle 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 {\displaystyle 1/x} is positive but less than the rational function 1 {\displaystyle 1} . In fact, if n {\displaystyle n} is any natural number, then n ( 1 / x ) = n / x {\displaystyle n(1/x)=n/x} is positive but still less than 1 {\displaystyle 1} , no matter how big n {\displaystyle n} is. Therefore, 1 / x {\displaystyle 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 y {\displaystyle y} , produces an example with a different order type.
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.
Every linearly ordered field K {\displaystyle K} contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 {\displaystyle 1} of K {\displaystyle 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 {\displaystyle K} . The following are equivalent characterizations of Archimedean fields in terms of these substructures.