TABLE OF CONTENTS In this chapter, we will provide in detail a rigorous treatment of
the mathematics behind the construction of hypermodels and
the mathematical study of hypermodels i.e. hyperanalysis. [*]
This chapter can be read independently. We will focus on aspects of hyperanalysis which are relevant to stochastic analysis and mathematical finance.
We assume in this part of the book that the reader has a solid background in calculus and has some exposure to abstract mathematics (including some logic and set theory). While this pure mathematics part of the book may not concern most practitioners, it is nevertheless recommended to the serious researchers in hypermodel applications as well as to the mathematically inquisitive readers.
In 7.1, we begin by developing the first order formulas and sentences (statements), then we define some basic objects in hyperanalysis such as hypermodels of 
these can be regarded as enriched models of the geometric line. The existence of these models is shown by two methods. The first one uses G del s completeness theorem. (Do not confuse this with the famous G del s incompleteness theorem, which concerns very different matters.) More precisely, we use the compactness theorem a general method which does not depend on specific set-theoretic constructions. The second one uses the ultrapower construction via a free ultrafilter; the fact that this method produces hypermodels requires Lo? theorem from logic. As a starter, we first concentrate on hypermodels from the ordered field structure of the real number system . Then we...
