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:
“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)
“Men are only as good as their technical development allows them to be.”
—George Orwell (1903–1950)
“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)