Editing Internal set theory (section)

{{Short description|System of mathematical set theory}}
'''Internal set theory''' ('''IST''') is a mathematical theory of [[Set (mathematics)|sets]] developed by [[Edward Nelson]] that provides an axiomatic basis for a portion of the [[nonstandard analysis]] introduced by [[Abraham Robinson]]. Instead of adding new elements to the [[real number]]s, 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  [[Zermelo–Fraenkel axioms|ZFC axioms for sets]].  Thus, IST is an enrichment of [[Zermelo–Fraenkel set theory|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.