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:
“This was the Eastham famous of late years for its camp- meetings, held in a grove near by, to which thousands flock from all parts of the Bay. We conjectured that the reason for the perhaps unusual, if not unhealthful development of the religious sentiment here, was the fact that a large portion of the population are women whose husbands and sons are either abroad on the sea, or else drowned, and there is nobody but they and the ministers left behind.”
—Henry David Thoreau (1817–1862)
“Information about child development enhances parents’ capacity to respond appropriately to their children. Informed parents are better equipped to problem-solve, more confident of their decisions, and more likely to respond sensitively to their children’s developmental needs.”
—L. P. Wandersman (20th century)
“Every new development for the last three centuries has brought men closer to a state of affairs in which absolutely nothing would be recognized in the whole world as possessing a claim to obedience except the authority of the State. The majority of people in Europe obey nothing else.”
—Simone Weil (1909–1943)