Internal set theory

Summary

Internal set theory, written IST, is an axiomatic form of nonstandard analysis introduced by Edward Nelson in 1977. It adds to Zermelo–Fraenkel set theory with choice a new unary predicate "standard" and three axiom schemes: idealization, standardization and transfer. Instead of building a larger number system, IST finds nonstandard elements, such as infinitesimal reals and unlimited natural numbers, inside the usual sets. Nelson proved that IST is a conservative extension of ZFC. So every statement in the language of ZFC that IST proves is already a theorem of ZFC.