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In general, the state of a system in which a physical quantity such as mass is distributed over space can be modeled by a function that represents the density distribution. The purpose of this paper is to introduce special functions that can be applied when in the system to be modeled, where the quantity is distributed over isolated points. For that matter, the expanded real numbers are introduced as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and hyperreal delta functions are defined as special functions from the real numbers to the expanded real numbers satisfying the condition that (i) the support is a singleton, and (ii) the integral over the reals is a nonzero real number. These newly defined hyperreal delta functions, and tensor products thereof, then provide a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

Cilj diplomskega dela je z uvedbo infinitezemalnih števil predstaviti drugačen pristop h klasični analizi. Infinitezimalna števila uvedemo s pomočjo petih aksiomov. Največji poudarek je na principu prenosa, ki povezuje trditve, ki veljajo nad hiperrealnimi števili, s trditvami, ki veljajo nad realnimi števili. Skozi diplomsko delo dokazujemo, da so definicije iz nestandardne analize ekvivalentne tistim iz klasične analize, nato pa nove definicije uporabimo za dokaze izrekov iz klasične analize. Zaradi obširnosti klasične analize, se v diplomskem delu omejimo na koncepte zveznosti, limite, odvoda in kompaktnosti. In the thesis we introduce infinitesimals to present a new approach to classical analysis. We do so by introducing a set of five axioms. The emphasis is on the transfer axiom which helps us transfer theorems from nonstandard analysis to classical analysis. Throughout the thesis we show that the definitions in non-standard analysis are equivalent to those from classical analysis, and use non-standard definitions to prove theorems from classical analysis. Due to the large reach of classical analysis, we limit ourselves to concepts of continuity, limits, differentiation and compactness.

We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's nonstandard analysis $^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ${^{\bullet}\mathbb{R}}$ and in ${^\ast\mathbb{R}}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of "non-nonstandard" analysis. In particular, we will show that there exists a nonzero element in Henle's ring that is "too small" to be registered as nonzero in Paolo Giordano's ring, while it is seen as a nonzero infinitesimal in ${^\ast\mathbb{R}}$. This result suggests that some hyperreal infinitesimals are smaller than the nilpotent infinitesimals. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in ${^{\bullet}\mathbb{R}}$.

There are many natural phenomena that can best be described by the use of infinitesimal and infinite numbers (see e.g. [ 1 , 5 , 13 , 23 ]. However, until now, the Non-standard techniques have been applied to theoretical models. In this paper we investigate the possibility to implement such models in numerical simulations. First we define the field of Euclidean numbers which is a particular field of hyperreal numbers. Then, we introduce a set of families of Euclidean numbers, that we have called altogether algorithmic numbers, some of which are inspired by the IEEE 754 standard for floating point numbers. In particular, we suggest three formats which are relevant from the hardware implementation point of view: the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the Truncated Algorithmic Numbers. In the second part of the paper, we show a few applications of such numbers.

International audience; In update logic as studied by Baltag, Moss, Solecki and van Benthem, little attention is paid to the interpretation of an action by an agent, which is just assumed to depend on the situation. This is actually a complex issue that nevertheless complies to some logical dynamics. In this paper, we tackle this topic. We also deal with actions that change propositional facts of the situation. In parallel, we propose a formalism to accurately represent an agent's epistemic state based on hyperreal numbers. In that respect, we use infinitesimals to express what would surprise the agents (and by how much) by contradicting their beliefs. We also use a subjective probability to model the notion of belief. It turns out that our probabilistic update mechanism satisfies the AGM postulates of belief revision.

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17th through the 20th centuries. We will also present `didactic observations' at relevant places in the historical account.

This paper first describes the construction and use of the hyperreals in the theorem-prover Isabelle within the framework of higher-order logic (HOL). The theory, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his nonstandard analysis (NSA). The construction of the hyperreal number system has been carried out strictly through the use of definitions to ensure that the foundations of NSA in Isabelle are sound. Mechanizing the construction has required that various number systems including the rationals and the reals be built up first. Moreover, to construct the hyperreals from the reals has required developing a theory of filters and ultrafilters and proving Zorn's lemma, an equivalent form of the axiom of choice.This paper also describes the use of the new types of numbers and new relations on them to formalize familiar concepts from analysis. The current work provides both standard and nonstandard definitions for the various notions, and proves their equivalence in each case. To achieve this aim, systematic methods, through which sets and functions are extended to the hyperreals, are developed in the framework. The merits of the nonstandard approach with respect to the practice of analysis and mechanical theorem-proving are highlighted throughout the exposition.

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

Kirchhoff's laws fail to hold in general for infinite electrical networks. Standard calculus is simply incapable of resolving this paradox because it cannot provide the infinitesimals and more generally the hyperreal currents and voltages that such networks often require. However, nonstandard analysis can do precisely this. The idea of a nonstandard electrical network is introduced in this paper and is used to reestablish Kirchhoff's laws for a fairly broad class of infinite electrical networks. The second section herein presents a fairly brief tutorial on infinitesimals, hyperreal numbers, and the key ideas of nonstandard analysis needed for a comprehension of this paper.

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

A nonstandard universe is constructed from a superstructure in a Boolean-valued model of set theory. This provides a new framework of nonstandard analysis with which methods of forcing are incorporated naturally. Various new principles in this framework are provided together with the following applications: (1) An example of an ℵ1-saturated Boolean ultrapower of the real number field which is not Scott complete is constructed. (2) Infinitesimal analysis based on the generic extension of the hyperreal numbers is provided, and the hull completeness theorem and the Loeb measure construction are extended to objects in the generic extension of the internal universe. (3) The reduction theory of the Boolean-valued complex numbers are developed as a prototype of the applications to the topological reduction theory of Boolean sheaves or operator algebras.

The infinitely small and the infinitely large – in one form or another – are essential in calculus. In fact, they are among the distinguishing features of calculus compared to some other branches of mathematics, for example algebra. They have appeared throughout the history of calculus in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the seventeenth through the twentieth centuries. This will, in fact, entail discussing central issues in the development of calculus.

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.