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:
“If you complain of people being shot down in the streets, of the absence of communication or social responsibility, of the rise of everyday violence which people have become accustomed to, and the dehumanization of feelings, then the ultimate development on an organized social level is the concentration camp.... The concentration camp is the final expression of human separateness and its ultimate consequence. It is organized abandonment.”
—Arthur Miller (b. 1915)
“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)
“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)