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.)
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“The man, or the boy, in his development is psychologically deterred from incorporating serving characteristics by an easily observable fact: there are already people around who are clearly meant to serve and they are girls and women. To perform the activities these people are doing is to risk being, and being thought of, and thinking of oneself, as a woman. This has been made a terrifying prospect and has been made to constitute a major threat to masculine identity.”
—Jean Baker Miller (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)
“Men are only as good as their technical development allows them to be.”
—George Orwell (1903–1950)