The purpose of this text is twofold. The first purpose is to develop basic nonstandard analysis by an axiomatic ap-proach. The axiom system we use is a modified version of an axiom system in-troduced by Henson; The second purpose is to define and use our superstructures within ZFC, withouturelements. To do this, we introduce a general type of hierarchy of sets, of whichsuperstructures as well as the von Neumann cumulative hierarchy are special cases.We develop a little theory about such hierarchies, and we define a superstructureas such a hierarchy, satisfying some extra conditions. It is these superstructureswe use to define nonstandard extensions of, using axioms. This is also done in fulltechnical detail. |