As is known, Dedekind completeness is one of the basic concepts of real analysis, which arises immediately when constructing the line of real numbers. Since this property has many applications in various situations, alternative properties equivalent to Dedekind completeness naturally arise. In this article, the main attention is focused on the description of such properties, on proving equivalence to completeness and on individual examples of applications. In particular, a modified definition of Dedekind cuts is introduced, which made it possible to classify them as principal and free cuts, which serve as convenient models of rational and irrational numbers, respectively. The axioms of Cantor and Archimedes and their connection with Dedekind completeness in ordered fields and in ordered sets are considered. A connection is found between the fulfillment of the Archimedes axiom and the presence of a countable everywhere dense set in ordered fields satisfying the Cantor axiom.

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