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.