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)
“Ultimately, it is the receiving of the child and hearing what he or she has to say that develops the child’s mind and personhood.... Parents who enter into a dialogue with their children, who draw out and respect their opinions, are more likely to have children whose intellectual and ethical development proceeds rapidly and surely.”
—Mary Field Belenky (20th century)
“The work of adult life is not easy. As in childhood, each step presents not only new tasks of development but requires a letting go of the techniques that worked before. With each passage some magic must be given up, some cherished illusion of safety and comfortably familiar sense of self must be cast off, to allow for the greater expansion of our distinctiveness.”
—Gail Sheehy (20th century)