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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“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)
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—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
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—Benjamin Harrison (1833–1901)