A survey of nonstandard set theory

Abstract

This survey paper presents an up-to-date account of Nonstandard Set Theory (NST). The introduction presents a brief historical perspective on motivation and the techniques exploited for the realisation of alternative models for the system, first systematically advanced in Robinson's work. It also elaborates on the need for an axiomatic foundation for nonstandard analysis as against an ultrapower enlargement of typestructure or cumulative structure over the systen of real numbers or the system of natural numbers, in particular. In section 1, a systematization of the axiomatic foundation for nonstandard analysis is presented. The axioms of Extension, Transfer, Saturation and the principle of Internality together with their consequences are discussed. Section 2 presents a systematization of NST as a conservative or nonconservative extension of the Zermelo-Fraenkel set theory with the axiom of choice. It elaborates on Nelson's internal set theory in Section 2.1, on Hrbácek's axiomatics in Section 2.2, on Kawai's system, along with Kinoshita's refinements, in section 2.3, and on Fletcher's stratified nonstandard set theory in section 2.4. Finally, the paper indicates the possibility of relating NST to Alternative Set Theory (AST) in so far as the latter is concerned with constructing ultrafilters using various types of automosphisms and endomorphisms. It also notes that introducing some restricted form of infinitesimal analysis not dependent on the explicit use of the transfer principle, as opposed to pursuing approaches heavily depending on set-theoretic sophistication, might yield good results.