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:
“If you complain of people being shot down in the streets, of the absence of communication or social responsibility, of the rise of everyday violence which people have become accustomed to, and the dehumanization of feelings, then the ultimate development on an organized social level is the concentration camp.... The concentration camp is the final expression of human separateness and its ultimate consequence. It is organized abandonment.”
—Arthur Miller (b. 1915)
“The proper aim of education is to promote significant learning. Significant learning entails development. Development means successively asking broader and deeper questions of the relationship between oneself and the world. This is as true for first graders as graduate students, for fledging artists as graying accountants.”
—Laurent A. Daloz (20th century)
“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)