Development
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. (Kanovei and Shelah have found a method that gives an explicit construction, at the cost of a significantly more complicated treatment.)
Read more about this topic: Hyperreal Number
Famous quotes containing the word development:
“The work of adult life is not easy. As in childhood, each step presents not only new tasks of development but requires a letting go of the techniques that worked before. With each passage some magic must be given up, some cherished illusion of safety and comfortably familiar sense of self must be cast off, to allow for the greater expansion of our distinctiveness.”
—Gail Sheehy (20th century)
“On fields all drenched with blood he made his record in war, abstained from lawless violence when left on the plantation, and received his freedom in peace with moderation. But he holds in this Republic the position of an alien race among a people impatient of a rival. And in the eyes of some it seems that no valor redeems him, no social advancement nor individual development wipes off the ban which clings to him.”
—Frances Ellen Watkins Harper (1825–1911)
“John B. Watson, the most influential child-rearing expert [of the 1920s], warned that doting mothers could retard the development of children,... Demonstrations of affection were therefore limited. “If you must, kiss them once on the forehead when they say goodnight. Shake hands with them in the morning.””
—Sylvia Ann Hewitt (20th century)