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 experience of a sense of guilt for wrong-doing is necessary for the development of self-control. The guilt feelings will later serve as a warning signal which the child can produce himself when an impulse to repeat the naughty act comes over him. When the child can produce his on warning signals, independent of the actual presence of the adult, he is on the way to developing a conscience.”
—Selma H. Fraiberg (20th century)
“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)
“The young women, what can they not learn, what can they not achieve, with Columbia University annex thrown open to them? In this great outlook for women’s broader intellectual development I see the great sunburst of the future.”
—M. E. W. Sherwood (1826–1903)