Internal Set Theory

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 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. This is not part of the formal theory, but is a pedagogical device that might help the student interpret the formalism. The essential distinction, similar to the concept of definable numbers, contrasts the finiteness of the domain of concepts that we can specify and discuss with the unbounded infinity of the set of numbers; compare finitism.

The term standard is therefore intuitively taken to correspond to some necessarily finite portion of "accessible" whole numbers. The argument can be applied to any infinite set of objects whatsoever – there are only so many elements that one can specify in finite time using a finite set of symbols and there are always those that lie beyond the limits of our patience and endurance, no matter how we persevere. We must admit to a profusion of non-standard elements — too large or too anonymous to grasp — within any infinite set.

...
Wikipedia