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:
“A defective voice will always preclude an artist from achieving the complete development of his art, however intelligent he may be.... The voice is an instrument which the artist must learn to use with suppleness and sureness, as if it were a limb.”
—Sarah Bernhardt (1845–1923)
“Good schools are schools for the development of the whole child. They seek to help children develop to their maximum their social powers and their intellectual powers, their emotional capacities, their physical powers.”
—James L. Hymes, Jr. (20th century)
“The Cairo conference ... is about a complicated web of education and employment, consumption and poverty, development and health care. It is also about whether governments will follow where women have so clearly led them, toward safe, simple and reliable choices in family planning. While Cairo crackles with conflict, in the homes of the world the orthodoxies have been duly heard, and roundly ignored.”
—Anna Quindlen (b. 1952)