 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 decreased by some infinitesimal amount to give a smaller one). Edward Nelson (1977) pioneered an alternative approach—Internal Set Theory—on which the basic language of mathematics is enriched in order to allow us to distinguish between standard and non-standard real numbers, as well as “internal” and “external” sets. On Nelson’s approach, infinitesimals are non-standard real numbers that are smaller in absolute value than any positive standard real numbers. “Internal sets” are those that can be defined in the basic language, and they behave just the same as standard sets of standard reals—for instance, bounded internal sets always have a least upper bound. But the set of all infinitesimals, just like the set of all standard real numbers, is an “external set” of the theory that can’t be defined within the language, and thus doesn’t necessarily have a least upper bound.