The Archimedean property applies to unbounded sets, specifically real numbers and rational numbers, which both satisfy this property despite the latter not adhering to the Axiom of Completeness. The discussion highlights that while the Archimedean property relies on the existence of upper bounds, unbounded sets can still exhibit this property. The proof of the Archimedean property is contingent upon the context of the elements being discussed, particularly in relation to their completeness.
PREREQUISITESMathematicians, students of real analysis, and anyone interested in the properties of number sets and their completeness.
torquerotates said:Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.