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Depending on the construction of the real numbers used, completeness may take the form of an axiom (the completeness axiom), or may be a theorem proven from the construction. There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness ( completeness as a metric space ).
The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. [2] It can be used to prove many of the fundamental results of real analysis , such as the intermediate value theorem , the Bolzano–Weierstrass theorem , the extreme value theorem , and the Heine ...
If is the set of rational numbers, viewed as a totally ordered set with the usual numerical order, then each element of the Dedekind–MacNeille completion of may be viewed as a Dedekind cut, and the Dedekind–MacNeille completion of is the total ordering on the real numbers, together with the two additional values .
Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property. The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first ...
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 ).
Tarski stated, without proof, that these axioms turn the relation < into a total ordering.The missing component was supplied in 2008 by Stefanie Ucsnay. [2]The axioms then imply that R is a linearly ordered abelian group under addition with distinguished positive element 1, and that this group is Dedekind-complete, divisible, and Archimedean.
The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order axiom schema of ...
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (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. However, many other interesting notions ...