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:
“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)
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“For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.”
—Kyle D. Pruett (20th century)