In 1960, Abraham Robinson
provided an answer following the first approach. The extended set is called the hyperreals
and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard
In 1977 Edward Nelson
provided an answer following the second approach. The extended axioms are IST, which stands either for Internal Set Theory
or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number
which is less, in absolute value, than any positive standard real number.
In 2006 Karel Hrbacek
developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels i.e, in the coarsest level there are no infinitesimals nor unlimited numbers. Infinitesimals are in a finer level
and there are also infinitesimals with respect to this new level and so on.