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 ArchimedesOn 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 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.

Read more about Archimedean Property:  History and Origin of The Name of The Archimedean Property, Definition For Linearly Ordered Groups, Definition For Normed Fields

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Ultraproduct - Łoś's Theorem - Examples
... theorem implies that *S has the same property ... Consider, however, the Archimedean property of the reals, which states that there is no real number x such that x > 1, x > 1 +1, x > 1 + 1 + 1.. ... Łoś's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic ...
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... The following are equivalent characterizations of Archimedean fields in terms of these substructures ... case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound ... a situation that points up both the incompleteness and disconnectedness of any non-Archimedean field ...

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