Editing Transfer principle (section)

== Differences between R and <sup>*</sup>R==

The transfer principle however doesn't mean that '''R''' and *'''R''' have identical behavior. For instance, in *'''R''' there exists an element ''&omega;'' such that

: <math> 1<\omega, \quad 1+1<\omega, \quad 1+1+1<\omega, \quad 1+1+1+1<\omega, \ldots </math>

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 ''&omega;'' 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]].