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In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty sets ''A'' and ''B'', such that all elements of ''A'' are less than all elements of ''B'', and ''A'' contains no greatest element. Dedekind cuts are one method of construction of the real numbers. The set ''B'' may or may not have a smallest element among the rationals. If ''B'' has a smallest element among the rationals, the ''cut'' corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between ''A'' and ''B''. In other words, ''A'' contains every rational number ''less than'' the cut, and ''B'' contains every rational number ''greater than or equal to'' the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals. More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts ''A'' and ''B'', such that ''A'' is closed downwards (meaning that for all ''a'' in ''A'', ''x'' ≤ ''a'' implies that ''x'' is in ''A'' as well) and ''B'' is closed upwards, and ''A'' contains no greatest element. See also completeness (order theory). It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the ''B'' set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps. Dedekind used the German word ''ドイツ語:Schnitt'' (cut) in a visual sense rooted in Euclidean geometry. His theorem asserting the completeness of the real number system is nevertheless a theorem about numbers and not geometry. Classical Euclidean geometry lacked a treatment of continuity (although Eudoxus did construct a sophisticated theory of incommensurable quantities such as ): thus the very first proposition of the very first book of Euclid's geometry (constructing an equilateral triangle) was criticised by Pappus of Alexandria on the grounds that there was nothing in the axioms that asserted two intersecting circles in fact intersect in points. In David Hilbert's axiom system, continuity is provided by the Axiom of Archimedes, while in Alfred Tarski's system continuity is provided by what is essentially Dedekind's section. In mathematical logic, the identification of the real numbers with the real number line is provided by the Cantor–Dedekind axiom. == Representations == It is more symmetrical to use the (''A'',''B'') notation for Dedekind cuts, but each of ''A'' and ''B'' does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward closed set ''A'' without greatest element a "Dedekind cut". If the ordered set ''S'' is complete, then, for every Dedekind cut (''A'', ''B'') of ''S'', the set ''B'' must have a minimal element ''b'', hence we must have that ''A'' is the interval ( −∞, ''b'' In this case, we say that ''b'' ''is represented by'' the cut (''A'',''B''). The important purpose of the Dedekind cut is to work with number sets that are ''not'' complete. The cut itself can represent a number not in the original collection of numbers (most often rational numbers). The cut can represent a number ''b'', even though the numbers contained in the two sets ''A'' and ''B'' do not actually include the number ''b'' that their cut represents. For example if ''A'' and ''B'' only contain rational numbers, they can still be cut at √2 by putting every negative rational number in ''A'', along with every non-negative number whose square is less than 2; similarly ''B'' would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for √2, if the rational numbers are partitioned into ''A'' and ''B'' this way, the partition itself represents an irrational number. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dedekind cut」の詳細全文を読む スポンサード リンク
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