 As a result of the rigorous definitions in the calculus mentioned above, from the mid-19th century most mathematicians working in analysis abandoned infinitesimals. However, in the mid-20th century, Robinson (1966) showed that it is possible to give a rigorous definition of infinitesimals, and that infinitesimals can be used in a non-standard development of real analysis (D. Laugwitz also did similar work around the same time, but Robinson’s system has been more widely discussed). While he developed his non-standard analysis using model theory, subsequent developments have also been grounded in algebra and topology. Robinson’s approach supplies an extended number system—the hyperreal number system—that contains the standard real number system, plus further ‘infinitesimal’ numbers whose absolute values are greater than 0, but less than any positive standard real number. Robinson’s construction of the hyperreals provides a set with the same cardinality as the standard reals. Simple modifications of the construction can create sets of hyperreals of larger cardinality.