TABLE OF CONTENTS By classical hyperanalysis we mean the study of calculus properties of the real number system 
using hypermodels of . In this section, we provide this study a rigorous foundation via methods and techniques from mathematical logic. We then construct hypermodels for i.e. hyperreal number systems.
The arithmetical structure of real numbers forms what algebraists called a field, a structure in which addition, subtraction, multiplication and division obey the usual rules. If the structure has an ordering on it which satisfies the usual properties, we call it an ordered field. There are structures other than that satisfy the same rules. The ordered field theoretic properties of can be described using the language L which has symbols like:
addition symbol +
multiplication symbol
inequality symbol <
constant symbols 0 and 1.
Here + and are binary function symbols ( binary means taking two argument inputs), < is a binary relation symbol and 0, 1 are constant symbols but can be conveniently regarded as relation symbols that takes no argument inputs. Note that subtraction and division symbols are not included, as we shall see that these are just inverse operations they can be defined from the corresponding basic operations. The significance of the ordering lies in the fact that we can use it to define positive and negative.
The above list represents symbols that...
