Differences Between R and *R
The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that
but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like ω is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.
The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.
Read more about this topic: Transfer Principle
Famous quotes containing the word differences:
“I don’t know what immutable differences exist between men and women apart from differences in their genitals; perhaps there are some other unchangeable differences; probably there are a number of irrelevant differences. But it is clear that until social expectations for men and women are equal, until we provide equal respect for both men and women, our answers to this question will simply reflect our prejudices.”
—Naomi Weisstein (b. 1939)