For the American record producer known professionally as Dedekind Cut, see Fred Warmsley.
Dedekind used his cut to construct the irrational, real numbers.
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand[1][2]), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two sets A and B, such that each element of A is less than every element of B, and A contains no greatest element. 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.[3] 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.[3]
Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as 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.
^Bertrand, Joseph (1849). Traité d'Arithmétique. page 203. An incommensurable number can be defined only by indicating how the magnitude it expresses can be formed by means of unity. In what follows, we suppose that this definition consists of indicating which are the commensurable numbers smaller or larger than it ....
^Spalt, Detlef (2019). Eine kurze Geschichte der Analysis. Springer. doi:10.1007/978-3-662-57816-2. ISBN 978-3-662-57815-5.
^ abDedekind, Richard (1872). Continuity and Irrational Numbers(PDF). Section IV. Whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....
In mathematics, Dedekindcuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction...
contribution is the definition of real numbers through the notion of Dedekindcut. He is also considered a pioneer in the development of modern set theory...
Alternate/Endings. His debut studio album was released under the name DedekindCut. The entitled $uccessor, was released in November 2016.[unreliable source...
real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts...
as Dedekindcuts of rational numbers. For convenience we may take the lower set A {\displaystyle A\,} as the representative of any given Dedekindcut (...
include equivalence classes of Cauchy sequences (of rational numbers), Dedekindcuts, and infinite decimal representations. All these definitions satisfy...
set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a Dedekindcut of rationals: a non-empty...
two-sided set. By insisting that L<R, this two-sided set resembles the Dedekindcut. The resulting construction yields a field, now called the surreal numbers...
an American record producer and disc jockey known professionally as DedekindCut (formerly Lee Bannon). It includes a list of songs produced, co-produced...
axiom Dedekind completeness DedekindcutDedekind discriminant theorem Dedekind domain Dedekind eta function Dedekind function Dedekind group Dedekind number...
equivalent definition of computable numbers via computable Dedekindcuts. A computable Dedekindcut is a computable function D {\displaystyle D\;} which when...
(annotated index) – Annotated index of various modes of convergence Dedekindcut – Method of construction of the real numbers Lang 1992. Ebbinghaus, Heinz-Dieter...
constructed from the rational numbers by completion, using Cauchy sequences, Dedekindcuts, or infinite decimals (see Construction of the real numbers). The term...
complex numbers and outlines their properties. (In the third edition, the Dedekindcut construction is sent to an appendix for pedagogical reasons.) Chapter...
is definable in the language of arithmetic (or arithmetical) if its Dedekindcut can be defined as a predicate in that language; that is, if there is...
is the set of all dyadic rationals greater than a (reminiscent of a Dedekindcut). Thus the real numbers are also embedded within the surreals. There...
mathematical logic, the Peano axioms (/piˈɑːnoʊ/, [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers...
included the remix by Juliana Huxtable.) Black History Month in 3D Mix with DedekindCut fka Lee Bannon 2016 LGBT culture in New York City List of LGBT people...
Successor (EP), an EP by Sonata Arctica Successor (album), an album by DedekindCut A successor cardinal A successor ordinal The successor function, the...
involved. His method anticipated that of the Dedekindcut in the modern definition of real numbers by Richard Dedekind (1831–1916). In the Posterior Analytics...
independence of Euclid's fifth postulate. 1872 – Richard Dedekind invents what is now called the DedekindCut for defining irrational numbers, and now used for...
began to yield to the discrete with mathematician Richard Dedekind's (1831–1916) Dedekindcut, and Ludwig Boltzmann's (1844–1906) statistical thermodynamics...
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