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:
“For the child whose impulsiveness is indulged, who retains his primitive-discharge mechanisms, is not only an ill-behaved child but a child whose intellectual development is slowed down. No matter how well he is endowed intellectually, if direct action and immediate gratification are the guiding principles of his behavior, there will be less incentive to develop the higher mental processes, to reason, to employ the imagination creatively. . . .”
—Selma H. Fraiberg (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)
“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)