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 man, or the boy, in his development is psychologically deterred from incorporating serving characteristics by an easily observable fact: there are already people around who are clearly meant to serve and they are girls and women. To perform the activities these people are doing is to risk being, and being thought of, and thinking of oneself, as a woman. This has been made a terrifying prospect and has been made to constitute a major threat to masculine identity.”
—Jean Baker Miller (20th century)
“I have an intense personal interest in making the use of American capital in the development of China an instrument for the promotion of the welfare of China, and an increase in her material prosperity without entanglements or creating embarrassment affecting the growth of her independent political power, and the preservation of her territorial integrity.”
—William Howard Taft (1857–1930)
“Theories of child development and guidelines for parents are not cast in stone. They are constantly changing and adapting to new information and new pressures. There is no “right” way, just as there are no magic incantations that will always painlessly resolve a child’s problems.”
—Lawrence Kutner (20th century)