The hyperreal line (the inflate of the reals) is an ordered field

Now suppose that the set \(\bbN\) of natural numbers is a member of \(U\). Then so is the set \(\Re\) of real numbers, since each real number may be identified with a set of natural numbers. \(\Re\) may be regarded as an ordered field, and the same is therefore true of its inflate \(\hat{\Re}\), since the latter has precisely the same first-order properties as \(\Re\). \(\hat{\Re}\) is called the hyperreal line, and its members hyperreals. A standard hyperreal is then just a real, to which we sh