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Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers , whose corresponding number line has a "gap" at each irrational value.
All completeness properties are described along a similar scheme: one describes a certain class of subsets of a partially ordered set that are required to have a supremum or required to have an infimum. Hence every completeness property has its dual, obtained by inverting the order-dependent definitions in the given statement. Some of the ...
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
Completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval (0,1), which is not complete.
Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable; Mathematics ... Completeness, a 1998 ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
Dedekind completeness implies other sorts of completeness (see below), but also has some important consequences. Archimedean property : for every real number x , there is an integer n such that x < n {\displaystyle x<n} (take, n = u + 1 , {\displaystyle n=u+1,} where u {\displaystyle u} is the least upper bound of the integers less than x ).
A formal system S is syntactically complete or deductively complete or maximally complete or negation complete if for each sentence (closed formula) φ of the language of the system either φ or ¬φ is a theorem of S. Syntactical completeness is a stronger property than semantic completeness.