Edward Nelson's Internal Set Theory enriches the basic language of mathematics to allow distinction between standard and non-standard real numbers.

For results stated in a first-order logical language, the hyperreals and the standard reals satisfy the transfer principle. But for results about sets, they behave differently. Every bounded set of standard reals has a least upper bound. However, for instance, the set of infinitesimal hyperreals is bounded (every member is less than .00001, among other bounds), but there is no least upper bound (no infinitesimal is an upper bound for all of the others, and every finitely large upper bound can be