Concept
Archimedean property
Overview
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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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In union with itself, it does not by its nature produce anything imperfect or foreign. All its changes are defined; it either produces itself or the Decad, either the homogeneous or the perfect.
Plutarch, Selected Essays of Plutarch, Vol. II.
States what Archimedean property is.
Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid--corpus solidum--and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.
Friedrich Engels, Landmarks of Scientific Socialism
States what Archimedean property is.
Take such a property as being named Jones. If a man is named Jones, so is his son; we will therefore call the property of being called Jones hereditary with respect to the relation of father and son. If a man is called Jones, all his descendants in the direct male line are called Jones; this follows from the fact that the property is hereditary. Now, instead of the relation of father and son, consider the relation…
Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy
States what Archimedean property is.
Related ideas
- Infinitesimal
- Archimedes
- Real number
- Natural number
- Reductio ad absurdum
- Euclid's Elements
- Infimum and supremum
- Eudoxus of Cnidus
- Mathematical analysis
- Heuristic
- Ancient Greece
- Mathematical proof
- David Hilbert
- Ordered field
Place in the tradition
Part of Second-order logic.