Dedekind cut

Dedekind cut
two nonempty subsets of an ordered field, as the rational numbers, such that one subset is the collection of upper bounds of the second and the second is the collection of lower bounds of the first: can be used to define the real numbers in terms of the rational numbers.
[named after J.W.R. DEDEKIND]

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      in mathematics, concept advanced in 1872 by the German mathematician Richard Dedekind (Dedekind, Richard) that combines an arithmetic formulation of the idea of continuity with a rigorous distinction between rational and irrational numbers (irrational number). Dedekind reasoned that the real numbers (real number) form an ordered continuum, so that any two numbers x and y must satisfy one and only one of the conditions x < y, x = y, or x > y. He postulated a cut that separates the continuum into two subsets, say X and Y, such that if x is any member of X and y is any member of Y, then x < y. If the cut is made so that X has a largest rational member or Y a least member, then the cut corresponds to a rational number. If, however, the cut is made so that X has no largest rational member and Y no least rational member, then the cut corresponds to an irrational number.

      For example, if X is the set of all real numbers x less than or equal to 22/7 and Y is the set of real numbers y greater than 22/7, then the largest member of X is the rational number 22/7. If, however, X is the set of all real numbers x such that x2 is less than or equal to 2 and Y is the set of real numbers y such that y2 is greater than 2, then X has no largest rational member and Y has no least rational member: the cut defines the irrational number √2.

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Universalium. 2010.

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