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This research discusses the validity of the properties of real numbers set to hyperreal numbers set, i.e. algebraic, ordered, and completeness properties, by using a finitely additive measure. This finitely additive measure is a map from the power set of natural numbers set onto set {0,1}. The subset of natural numbers set has measure zero if it’s finite and one if it’s infinite. The set of hyperreal numbers is constructed from equivalence classes of the set of all sequences of real numbers by using a relation involving the finitely additive measure, that is, two sequences of real numbers are said to be related if and only if those two sequences are the same almost everywhere. In the hyperreal numbers set, there exist infinitesimal numbers besides 0. Infinitesimal number is a number which is less than any positive real number and greater than any negative real number. So, in hyperreal number set, there are some smallest positive numbers. The results show that the hyperreal numbers set is an ordered and complete field.

Cardinality arguments against regular probability measures aim to show that no matter which ordered field H we select as the measures for probability, we can find some event space F of sufficiently large cardinality such that there can be no regular probability measure from F into H. In particular, taking H to be hyperreal numbers won’t help to guarantee that probability measures can always be regular. I argue that such cardinality arguments fail, since they rely on the wrong conception of the role of numbers as measures of probability. With the proper conception of their role we can see that for any event space F, of any cardinality, there are regular hyperreal-valued probability measures.

I mplicit functions arise throughout mathematical analysis. They are especially prominent in applications to economics. From the beginning of mathematical economics one must work with implicit functions, as in analysis of a consumer and his indifference curves, or of a firm and its isoquants. Consequently, in teaching calculus to business students we come early to the challenge of explaining differentiation of implicit functions. The aim of this note is to give a systematic approach to computing the successive derivatives of an implicit function of one real variable, an approach that we think brings students naturally to the main results. The idea is that smooth curves looked at closely enough are straight, so that analysis problems are locally problems of linear algebra. Our method invokes that intuition: we look at curves through (virtual) microscopes—but we pay for this by having to deal with hyperreal numbers. Although Leibniz and Newton, for instance, already worked with ‘‘infinitesimals’’, the rigorous treatment called ‘‘nonstandard analysis’’ was introduced only in 1961 by A. Robinson [4]. An especially simple presentation for didactical purposes was given by Keisler [3]; we shall mostly adopt his definitions and notations. We recall that the hyperreal numbers extend the real ones with the same algebraic rules; technically, the set *R of the hyperreal numbers is a non-archimedean ordered field in which the real line R is embedded. Moreover, *R contains at least one infinitesimal (and then it must contain infinitely many). An infinitesimal is a number e such that its absolute value is less thanevery real number butwhich is unequal to 0; its reciprocal 1e is infinite, that is, is a number whose absolute value is greater than every real number. Clearly, non-zero infinitesimals and infinite numbers are not real. A hyperreal number x, which is not infinite, is of course said to be finite; for any such x there exists one and only one real number r, which is infinitely close to x, that is, such that the difference x r is an infinitesimal: this r is called the standard part of x and is denoted by r = st(x). Formally, st is a ring homomorphism from the set of finite hyperreal numbers to R; and its kernel is the set of infinitesimals. Moreover, a function of one or several real variables may have a natural extension to the hyperreal numbers, with the same definition and the same properties as the original one. Indeed, if a real-valued function is defined by a system of formulas, its extension can be obtained by applying the same formulas to the hyperreal system. In this article, we adopt the same notation for a real function and for its natural extension. The concept of (virtual) microscope is well known (see, for example, [1, 2, 5]). For a point P(a, b) in the hyperreal plane *R and a positive infinite hyperreal number H, a microscope pointed on P and with H as power magnifies the distances from P by a factor H; more explicitly, it is a map, denoted by MPH ; defined on *R as follows: MPH : x; y ð Þ 7! X ;Y ð Þ with X 1⁄4 H x a ð Þ and Y 1⁄4 H y b ð Þ

Chapter 2 discusses different systems of numbers, emphasizing on reasons making them so important. It starts with the system of natural numbers and presents its existence as a natural consequence from the axiom of infinity. Next, the systems of integers and rational numbers are constructed over the system of natural numbers. The least upper bound property and its lack in the system of rational numbers are emphasized. The system of real numbers is introduced as an ordered field with least upper bound property, which improves the system of rational numbers. For completeness of the whole picture on numbers, brief information about the systems of extended real, complex and hyperreal numbers is also given. The chapter ends with a discussion of cardinality, emphasizing on finite, countable and continuum cardinalities.

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter)., 17 pages, 1 figure

In ordinary qualitative reasoning (QR), qualitative behavior of the dynamical systems is predicted by assignment of qualitative values such as \{+, 0, $-$\} into model variables based on proper transition rules. Unfortunately, due to ambiguities in these ordinarily introduced qualitative values, their arithmetic and transition rules cause predictions to be redundant, sometimes even inacurate. In this paper, we present a new method of qualitative reasoning which, besides using hyperreal numbers, takes into account their $\varepsilon$-{\bf H} ranking in describing both qualitative values and qualitative derivatives of variables and also employs a convergence filter to investigate the infinitesimal asymptotic behavior of qualitative variables. We applied this qualitative reasoning method to envision a temporally hierarchical complex system. The result shows that this method provides a more detailed and natural qualitative solution than previous methods like Kuipers's time abstraction in envisioning temporally hierarchical complex system.