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)
“The proper aim of education is to promote significant learning. Significant learning entails development. Development means successively asking broader and deeper questions of the relationship between oneself and the world. This is as true for first graders as graduate students, for fledging artists as graying accountants.”
—Laurent A. Daloz (20th century)
“As a final instance of the force of limitations in the development of concentration, I must mention that beautiful creature, Helen Keller, whom I have known for these many years. I am filled with wonder of her knowledge, acquired because shut out from all distraction. If I could have been deaf, dumb, and blind I also might have arrived at something.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)