Transfer Principle - Differences Between R and *R

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.