Dedekind's axiom - définition. Qu'est-ce que Dedekind's axiom
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Qu'est-ce (qui) est Dedekind's axiom - définition

METHOD OF CONSTRUCTION OF THE REAL NUMBERS
Dedekind cuts; Dedekind section; Completion (order theory); Dedekind's Axiom; Dedekind Cut
  • irrational]], [[real number]]s.

Dedekind cut         
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, 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 all elements of A are less than all elements of B, and A contains no greatest element.
Axiom schema         
A FORMULA IN THE METALANGUAGE OF AN AXIOMATIC SYSTEM IN WHICH ONE OR MORE SCHEMATIC VARIABLES APPEAR
Axiom scheme; Axiom schemata; Axiom-scheme; Finite axiomatization
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
Axiom of extensionality         
AXIOM OF ZERMELO–FRAENKEL SET THEORY ASSERTING THAT SET EQUALITY IS DETERMINED BY THE MEMBERSHIP RELATION
Axiom of extension; Axiom of Extensionality; Axiom extensionality; Extensionality axiom; Axiom of equality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set.

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Dedekind cut

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, 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 all elements of A are less than all elements 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. 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.

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, xa 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.