Di Nasso, Mauro On the foundations of nonstandard mathematics. (English) Zbl 0937.03075 Math. Jap. 50, No. 1, 131-160 (1999). The author presents a survey of some but not all of the various foundational approaches to nonstandard analysis. His main concern is with various axiomatic approaches relative to ZFC or ZFC without regularity. In part, he discusses the superstructure approach, and the axiomatic theories: Nelson’s Internal Set Theory IST, Hrbáček’s Nonstandard Theories, Kawai’s System NST, Fletcher’s Stratified Nonstandard System SNST, Ballard’s Enlargement Set Theory EST and others. These axiomatic approaches are intended to improve upon the superstructure approach by not insisting upon atoms (and using a set-theory such as ZFA + AC + A (infinite)) or base sets as a foundation, and eliminating what some may perceive as undesirable limitations associated with the “finite” rank approach and restricted modes of expression. I am not convinced that such improvements are altogether necessary and are, indeed, significant relative to applications. Reviewer: R.A.Herrmann (Annapolis) Cited in 3 Documents MSC: 03Hxx Nonstandard models 03E70 Nonclassical and second-order set theories Keywords:superstructures; axiomatic nonstandard analysis; survey PDFBibTeX XMLCite \textit{M. Di Nasso}, Math. Japon. 50, No. 1, 131--160 (1999; Zbl 0937.03075)