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, 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 infinitesimal elements.
Read more about Internal Set Theory: Intuitive Justification, Formal Axioms For IST, Formal Justification For The Axioms
Famous quotes containing the words internal, set and/or theory:
“Personal change, growth, development, identity formation—these tasks that once were thought to belong to childhood and adolescence alone now are recognized as part of adult life as well. Gone is the belief that adulthood is, or ought to be, a time of internal peace and comfort, that growing pains belong only to the young; gone the belief that these are marker events—a job, a mate, a child—through which we will pass into a life of relative ease.”
—Lillian Breslow Rubin (20th century)
“I can add colors to the chameleon,
Change shapes with Proteus for advantages,
And set the murderous Machiavel to school.”
—William Shakespeare (1564–1616)
“The theory seems to be that so long as a man is a failure he is one of God’s chillun, but that as soon as he has any luck he owes it to the Devil.”
—H.L. (Henry Lewis)