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.