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.)
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Famous quotes containing the word development:
“To be sure, we have inherited abilities, but our development we owe to thousands of influences coming from the world around us from which we appropriate what we can and what is suitable to us.”
—Johann Wolfgang Von Goethe (1749–1832)
“The development of civilization and industry in general has always shown itself so active in the destruction of forests that everything that has been done for their conservation and production is completely insignificant in comparison.”
—Karl Marx (1818–1883)
“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)