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:
“Sleep hath its own world,
And a wide realm of wild reality.
And dreams in their development have breath,
And tears, and tortures, and the touch of joy.”
—George Gordon Noel Byron (1788–1824)
“As a final instance of the force of limitations in the development of concentration, I must mention that beautiful creature, Helen Keller, whom I have known for these many years. I am filled with wonder of her knowledge, acquired because shut out from all distraction. If I could have been deaf, dumb, and blind I also might have arrived at something.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Dissonance between family and school, therefore, is not only inevitable in a changing society; it also helps to make children more malleable and responsive to a changing world. By the same token, one could say that absolute homogeneity between family and school would reflect a static, authoritarian society and discourage creative, adaptive development in children.”
—Sara Lawrence Lightfoot (20th century)