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:
“I’ve always been impressed by the different paths babies take in their physical development on the way to walking. It’s rare to see a behavior that starts out with such wide natural variation, yet becomes so uniform after only a few months.”
—Lawrence Kutner (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.”
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“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)