In 1960, Abraham Robinson established the framework in which the
existence of
infinitesimals is made precise. He showed that we may construct
enlargements of ultrapowers in such a way
that a mathematical structure contained in the ultrapower is embedded in
the enlargement, and so that true statements about the embedded structure
are true for the enlarged structure, and visa versa (with suitable
interpretation). This construction, when applied to the real numbers,
results in an enlarged totally ordered field containing infinite as well
as infinitesimals along with finite real numbers.
