Completeness of The Real Numbers - Forms of Completeness - Dedekind Completeness

Dedekind Completeness

See Dedekind completeness for more general concepts bearing this name.

Dedekind completeness is the property that every Dedekind cut of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom.

The rational number line Q is not Dedekind complete. An example is the Dedekind cut

L does not have a maximum and R does not have a minimum, so this cut is not generated by a rational number.

There is a construction of the real numbers based on the idea of using Dedekind cuts of rational numbers to name real numbers; e.g. the cut (L,R) described above would name . If one were to repeat the construction with Dedekind cuts of real numbers, one would obtain no additional numbers because the real numbers are Dedekind complete.

Read more about this topic:  Completeness Of The Real Numbers, Forms of Completeness

Other articles related to "dedekind, completeness":

Real Number - Properties - "The Complete Ordered Field"
... Additionally, an order can be Dedekind-complete, as defined in the section Axioms ... This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and ... These two notions of completeness ignore the field structure ...

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