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:
“As long as fathers rule but do not nurture, as long as mothers nurture but do not rule, the conditions favoring the development of father-daughter incest will prevail.”
—Judith Lewis Herman (b. 1942)
“America is a country that seems forever to be toddler or teenager, at those two stages of human development characterized by conflict between autonomy and security.”
—Anna Quindlen (b. 1952)
“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)