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)
“Understanding child development takes the emphasis away from the child’s character—looking at the child as good or bad. The emphasis is put on behavior as communication. Discipline is thus seen as problem-solving. The child is helped to learn a more acceptable manner of communication.”
—Ellen Galinsky (20th century)
“And then ... he flung open the door of my compartment, and ushered in “Ma young and lovely lady!” I muttered to myself with some bitterness. “And this is, of course, the opening scene of Vol. I. She is the Heroine. And I am one of those subordinate characters that only turn up when needed for the development of her destiny, and whose final appearance is outside the church, waiting to greet the Happy Pair!””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)