Nonstandard Methods of Analysis

Academia.edu, 2022

Re: the Continuum Hypothesis: Is there any set which has more members than the set of natural numbers (N), but fewer members than the set of real numbers (R)? The short answer is no. The long answer, that includes mathematical proof, shall be set forth in this paper.

Expositiones Mathematicae, 2003

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite "ideal" number α. The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.

arXiv: Logic, 2020

This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and infinitesimals. The construction of this new set is done naively in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its naivety character, the set $\mathbb{R}^{\mathbb{Z}_< }$ is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years.

Cantor’s [1878] Continuum Hypothesis (CH) is the Great Sphinx of mathematics. Gödel, who disavowed the CH, proved in 1940 that it is compatible with the ZFC axiomatic model. Cohen [1963, 1964] showed that rejection of the CH is equally compatible. This apparent indeterminacy invites some axiom conforming to intuition which answers the question with finality. In this note it is shown that the CH is inconsistent with a ubiquitous axiom of topology, namely, that the countable ordinals include as their supremum the first uncountable ordinal—i.e., that the countable ordinals are uncountable! It is hoped that this construction assists the progress of research on foundations.

2009

In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demonstrate how theorems in standard analysis “transfer over” to nonstandard analysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis.

The principal set-theoretic credos of nonstandard analysis are presented. A "naive" justification of the infinitesimal techniques and an overview of the corresponding formal apparatus are provided. The axioms of Nelson's internal set theory are discussed as well as those of the external set theories by Hrbaček and Kawai.

Mathematical Logic Quarterly, 1970

Using Alexander Yessenin-Volpin's theory of constructions in which the natural numbers are considered as an unfolding process over time, it is possible to develop a purely finitistic version of Real Analysis using only rational numbers. By employing an Intuitionistic Tense Logic and a parameter z for a “large” natural number, infinitesimals can be introduced and the theory of Real Analysis developed along the lines of A. Robinson’s Non Standard Analysis [Robinson, 1966]. The notion of a “very large” natural number is defined relative to a particular finite set S of proofs and calculations, so that z represents a natural number that has not yet arrived and is larger than any integer needed for the completion of the proofs and calculations in S. The notion “very large” is made precise axiomatically and in the Soundness Theorem of Section II of this paper. I have called the resulting theory Rational Constructive Analysis (RCA). Even though RCA borrows some of the strategies of Robinson in the use of the concept of infinitesimals, it is entirely constructive and essentially finitistic. It also bares a strong relation to Mycielski's theory FIN [Mycielski, 1981] which he has proposed as a possible finitization of Classical Analysis.

Journal of Applied and Industrial Mathematics, 2013

This is a biographical sketch and tribute to Abraham Robinson on the 95th anniversary of his birth with a short discussion of the place of nonstandard analysis in the present-day mathematics.

“Infinitesimal - A Dangerous Mathematical Theory” by Amir Alexander (Scientific American/Farrar – 2014) © A review essay by H. J. Spencer May 2017 This book is about one of the most dangerous ideas ever invented by human beings. It led to the theory that eventually destroyed the monopoly hold on the educated minds of Western Europe by planting one of the principal seeds of modern physics. Award-winning UCLA historian, Alexander has written a fascinating tale of how abstract ideas influence society. This book deserves to be read not just by scientists and engineers, whose lives are dominated by mathematics, but by any educated person who had to suffer through calculus class. This book is about one of the most dangerous ideas ever invented by human beings. It led to the theory that eventually destroyed the monopoly hold on the educated minds of Western Europe by planting one of the principal seeds of modern physics. Award-winning UCLA historian, Amir Alexander has written a fascinating tale of how abstract ideas influence society. This book deserves to be read not just by scientists and engineers, whose lives are dominated by mathematics, but by any educated person who had to suffer through calculus class. This book is much more than an esoteric history of an area of mathematics. It tracks the ancient rivalry between 'rationalists' and 'empiricists'. The dominant rationalists have always believed that human minds (at least those possessed by educated intellectuals) are capable of understanding the world purely by thought alone. The empiricists acknowledge that reality is far too complicated for humans to just guess its detailed structures. This is not simply an esoteric philosophical distinction but the difference in fundamental world-views that have deeply influenced the evolution of western civilization. In fact, rationalist intellectuals have usually looked to the logical perfection of mathematics as a justification for the preservation of religion and hierarchical social structures. In particular, the rationalists have raised the timeless, unchanging mathematical knowledge, represented by Euclidean geometry, as not just the only valid form of symbolic knowledge but as the only valid model of the logic of " proof ". In particular, this book focuses on the battle between the reactionaries (e.g. Jesuits and Hobbes), who needed a model of timeless perfection to preserve their class-based religious and social privileges and reality-driven modernists, like Galileo and Bacon, who were desirous of major changes. In the late Middle-Ages, the new order of Jesuits were the intellectual leaders of the Catholic Church and were formed to defeat the recent Reformation. They not only opposed Protestant theology but also the parallel forces of pluralism, populism and social reform. The Jesuits, like their Church itself and the ancient social structures they supported, were all organized on traditional (militaristic) hierarchical principles. In 1632, the Jesuits convened a major council in Rome and decided to ban the idea of " indivisibles " – the old idea that a line was composed of distinct and an infinity of tiny parts. They correctly anticipated that the threat of this idea to their rational view of the world, as an " orderly place, governed by a strict and unchanging set of rules. " Geometry was their best exemplar of their Catholic theology. The core of the disagreement was over the nature of the continuum, a concept that had surfaced in Ancient Greece. The reality of the idea of 'physical indivisibles' (atoms) was even being disputed by serious scientists as late as 1900. The original idea of continuity was stimulated by the apparent lack of observable 'gaps' in solids or liquid materials and our personal sense of the continuous flow of time (ideally, also endless). This idea became a key concept to many Ancient Greek thinkers; even, Aristotle made the plausible statement that: " No continuous thing is divisible into parts. " This was soon 'cast in concrete' with Euclid's major definition of a line as an infinite number of points. This became one of the core (obvious) assumptions of geometry – the basis of so much of western education. The concept of 'Continuity' became a key Principle of medieval scholastic thought, eagerly latched onto by Aquinas and other theologians in their battle with the atheistic and equally ancient idea of atoms.

Philosophia Mathematica, 2002

both recognized that there is something called 'the intuitive continuum'. It is the phenomenon of the intuitive continuum that motivates their developments of constructive real analysis on the basis of choice sequences. Brouwer already mentions the intuitive continuum and describes a few of its features in his doctoral thesis of 1907 [5]. Weyl, evidently unaware of Brouwer's early work, discusses the intuitive continuum in Chapter 2, section 6, of Das Kontinuum (DK) [27], which is entitled 'The Intuitive and the Mathematical Continuum'. The view of the intuitive continuum in DK is based largely on Husserlian phenomenological descriptions of the consciousness of internal time, although Weyl also mentions here and elsewhere some other historical views related to the idea of the intuitive continuum. 5 In this section of DK Weyl takes the experience of the flow of internal time as the model of the intuitive continuum. Brouwer and Weyl both distinguish 'internal', intuitive time from 'external' time. Brouwer calls the latter 'scientific' or measurable time [5, p.61]. Brouwer's notion of the primordial intuition of mathematics is concerned only with internal, intuitive time. Weyl, following Husserl, distinguishes 'phenomenal' time from 'objective' time and says that he will, in effect, 'suspend' the latter and focus only on the former [28, p.88]. It is the phenomenon of the intuitive continuum that he feels he has not captured with the mathematical (or 'arithmetic') theory of the continuum developed in DK. It is this phenomenon, in Weyl's eyes, to which Brouwer's development of intuitionistic real analysis does far more justice, and it is for just this reason that Weyl declares his allegiance to Brouwer in a famous paper published in 1921 [28]. We will describe below some of the features of the phenomenon on which Brouwer and Weyl are focused and then consider their mathematical treatments of it. A mathematics of the intuitive continuum should be founded on the formal or structural features of the intuitive continuum and, for Brouwer and Weyl, these will be structural features of the stream of consciousness that are 1 We are grateful to Robert Tragesser for discussion and comments.