An Introduction to General Nonstandard Analysis

In this chapter, we develop the general framework of nonstandard analysis and the necessary logic for the transfer principle. We will begin each section with a brief summary for readers who want to postpone the technical details until a later reading. The summary will note any important definitions and results of the section that the reader should know before going on. For example, Definition 2.1.1 describing a superstructure and Remark 2.1.3 are important in this section. A reader who wants quickly to get to later applications may skip Sects. 2.5, 2.7, and 2.9. The reader who has read the first chapter of this book will appreciate that Skolem functions will no longer be needed to replace the existential quantifier. The results obtained in the last chapter using our simple transfer principle will still be valid, since the transfer principle used here extends that simple one. The outline of this chapter is similar to that of Chap.2 of the author’s book with Albert E. Hurd, [4]. To work with general mathematical analysis, we need to consider sets, sets of sets, etc. All of these are constructed starting with a set of individuals. We think of an individual as an object different from a set. In particular, an individual contains no elements. We build our universe from the set X of individuals using the power set operation P . The set X will always contain the natural numbersN; usually it will contain R.