Archimedean property

An ordered group or field has the Archimedean property when, for every positive elements a and b, some natural-number multiple of a exceeds b. It excludes positive infinitesimals relative to any positive element.

Archimedean ordered group: An ordered group in which, for any positive elements a and b, some natural-number multiple of a exceeds b. This is the group-level setting in which the property is defined.
Ordered group: A group equipped with a total order preserved by translation. The property is defined for ordered groups as well as for fields.
Real number: A number in the complete ordered field extending the rational numbers. The real numbers satisfy the property, providing its standard model.
Archimedes: An ancient Greek mathematician who developed rigorous methods for comparing areas, volumes, and magnitudes. The property bears his name because it formalizes a principle associated with his comparisons of magnitudes.
Hölder's theorem: A theorem stating that every Archimedean linearly ordered group embeds order-preservingly into the additive group of real numbers. It turns the group-level property into a concrete representation by real numbers.
Archimedean ordered field: An ordered field in which every positive element is exceeded by some natural-number multiple of 1. In fields, testing multiples of 1 suffices to express the property.
Ordered field: A field with a total order compatible with addition and multiplication. The field version is a central setting for comparing real and nonstandard number systems.
Rational number: A number expressible as a ratio of integers with nonzero denominator. The rationals satisfy the property despite lacking completeness.
Archimedean axiom: An axiom asserting that repeated addition of a positive unit eventually exceeds any given magnitude. This classical formulation is the historical and geometric counterpart of the algebraic property.
Dedekind completeness: A property of a linear order in which every nonempty bounded-above subset has a least upper bound. The real numbers are both Archimedean and complete, but the rational numbers show completeness is not required.

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