forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space). The real numbers can be...

Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences. Archimedean property: for every real number...

completeness, complete, completed, or incompleteness in Wiktionary, the free dictionary. Complete may refer to: Completeness (logic) Completeness of a...

use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment. Cantor's construction of the real...

(poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices...

equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field...

constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. One of the standard illustrations...

number). The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset S {\displaystyle S} of the real numbers has an...

an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i...

the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers. More...

a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the...

Rational number Irrational number Completeness of the real numbers Least-upper-bound property Real line Extended real number line Dedekind cut 0 1 0.999...

known as the recursive numbers, effective numbers or the computable reals or recursive reals. The concept of a computable real number was introduced by...

(infinite) sets of elements. The basic property of the completeness of the real numbers that is required for defining and using real numbers involves a quantification...

critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least...

all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex...

theory T but true in the "standard" model of the natural numbers: φu is false in some other, "non-standard" models of T.) The completeness theorem makes a...

number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example:...

decode other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all...

sequence of rational numbers (e.g. via its decimal expansion, also see completeness of the real numbers). As another example, π is the limit of the sequence...

and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers Q because gaps exist...

real numbers such as the square root of 2 ( 2 ) {\displaystyle \left({\sqrt {2}}\right)} and π, and complex numbers which extend the real numbers with...

proof must use the completeness of the real numbers, which is not an algebraic property). This article describes the history of the theory of equations, called...

The following assertions are equivalent to ACA0 over RCA0: The sequential completeness of the real numbers (every bounded increasing sequence of real...

+\infty \right\}.} It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol + ∞ {\displaystyle...

This is a list of notable numbers and articles about notable numbers. The list does not contain all numbers in existence as most of the number sets are...

Real Analysis, Springer, p. 45, ISBN 9780817642112. Königsberger, Konrad (2003), "2.3 Die Vollständigkeit von R (the completeness of the real numbers)"...

mathematics, the irrational numbers (in- + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed...

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...

mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and...