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:
“The mother must teach her son how to respect and follow the rules. She must teach him how to compete successfully with the other boys. And she must teach him how to find a woman to take care of him and finish the job she began of training him how to live in a family. But no matter how good a job a woman does in teaching a boy how to be a man, he knows that she is not the real thing, and so he tends to exaggerate the differences between men and women that she embodies.”
—Frank Pittman (20th century)