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:
“Women, because of their colonial relationship to men, have to fight for their own independence. This fight for our own independence will lead to the growth and development of the revolutionary movement in this country. Only the independent woman can be truly effective in the larger revolutionary struggle.”
—Women’s Liberation Workshop, Students for a Democratic Society, Radical political/social activist organization. “Liberation of Women,” in New Left Notes (July 10, 1967)
“They [women] can use their abilities to support each other, even as they develop more effective and appropriate ways of dealing with power.... Women do not need to diminish other women ... [they] need the power to advance their own development, but they do not “need” the power to limit the development of others.”
—Jean Baker Miller (20th century)
“Every new development for the last three centuries has brought men closer to a state of affairs in which absolutely nothing would be recognized in the whole world as possessing a claim to obedience except the authority of the State. The majority of people in Europe obey nothing else.”
—Simone Weil (1909–1943)