Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment.
Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T. In particular, suitable nonstandard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements.
Nelson's formulation is made more accessible for the lay-mathematician by leaving out many of the complexities of meta-mathematical logic that were initially required to justify rigorously the consistency of number systems containing infinitesimal elements.
Whilst IST has a perfectly formal axiomatic scheme, described below, an intuitive justification of the meaning of the term standard is desirable.
Aside from the intuitive motivations suggested above, it is necessary to justify that additional IST axioms do not lead to errors or inconsistencies in reasoning.
Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work of Gottfried Leibniz, Johann Bernoulli, Leonhard Euler, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome[citation needed] real number-based arguments developed by Georg Cantor, Richard Dedekind, and Karl Weierstrass, which were perceived as being more rigorous by Weierstrass's followers.
This is quite similar to justifying the consistency of the axioms of elliptic non-Euclidean geometry by noting they can be modeled by an appropriate interpretation of great circles on a sphere in ordinary 3-space.