Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional axioms for sets. 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 non-standard elements within the set of real numbers can be shown to have properties that correspond to the properti