ABSTRACT
This chapter aims to offer an explanation from the point of view of logic, explaining how the principles used in the superstructure approach to nonstandard analysis are related to standard mathematical practice. One way to bring nonstandard analysis to bear in proofs by convergence is to use sequences indexed by the hyperintegers rather than the integers. Start from a part of mathematics, probability theory, where nonstandard methods have clearly offered new insights and enriched the field with new and interesting results. The chapter shows that rich spaces exist, a result that requires nonstandard analysis, and presented the theory of neometric spaces within the most general possible nonstandard framework, which we called the huge neometric family. It explains how the properties of internal sets in nonstandard hulls give rise to neocompact sets and how the saturation property of the nonstandard universe translates into countable compactness for the neometric family.
