- 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.
Other articles related to "dedekind, completeness":
... 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 ...
Famous quotes containing the word completeness:
“Poetry presents indivisible wholes of human consciousness, modified and ordered by the stringent requirements of form. Prose, aiming at a definite and concrete goal, generally suppresses everything inessential to its purpose; poetry, existing only to exhibit itself as an aesthetic object, aims only at completeness and perfection of form.”
—Richard Harter Fogle, U.S. critic, educator. The Imagery of Keats and Shelley, ch. 1, University of North Carolina Press (1949)