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:
“I can see ... only one safe rule for the historian: that he should recognize in the development of human destinies the play of the contingent and the unforeseen.”
—H.A.L. (Herbert Albert Laurens)
“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)
“Understanding child development takes the emphasis away from the child’s character—looking at the child as good or bad. The emphasis is put on behavior as communication. Discipline is thus seen as problem-solving. The child is helped to learn a more acceptable manner of communication.”
—Ellen Galinsky (20th century)