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
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“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)
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—Johann Wolfgang Von Goethe (1749–1832)
“The proper aim of education is to promote significant learning. Significant learning entails development. Development means successively asking broader and deeper questions of the relationship between oneself and the world. This is as true for first graders as graduate students, for fledging artists as graying accountants.”
—Laurent A. Daloz (20th century)