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:
“To be sure, we have inherited abilities, but our development we owe to thousands of influences coming from the world around us from which we appropriate what we can and what is suitable to us.”
—Johann Wolfgang Von Goethe (1749–1832)
“... work is only part of a man’s life; play, family, church, individual and group contacts, educational opportunities, the intelligent exercise of citizenship, all play a part in a well-rounded life. Workers are men and women with potentialities for mental and spiritual development as well as for physical health. We are paying the price today of having too long sidestepped all that this means to the mental, moral, and spiritual health of our nation.”
—Mary Barnett Gilson (1877–?)
“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)