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)
“For the child whose impulsiveness is indulged, who retains his primitive-discharge mechanisms, is not only an ill-behaved child but a child whose intellectual development is slowed down. No matter how well he is endowed intellectually, if direct action and immediate gratification are the guiding principles of his behavior, there will be less incentive to develop the higher mental processes, to reason, to employ the imagination creatively. . . .”
—Selma H. Fraiberg (20th century)
“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)