The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small∗
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An Introduction to Nonstandard Analysis 11
AN INTRODUCTION TO NONSTANDARD ANALYSIS ISAAC DAVIS Abstract. In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demon- strate how theorems in standard analysis \transfer over" to nonstandard anal- ysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis. 1. Introduction For many centuries, early mathematicians and physicists would solve problems by considering infinitesimally small pieces of a shape, or movement along a path by an infinitesimal amount. Archimedes derived the formula for the area of a circle by thinking of a circle as a polygon with infinitely many infinitesimal sides [1]. In particular, the construction of calculus was first motivated by this intuitive notion of infinitesimal change. G.W. Leibniz's derivation of calculus made extensive use of “infinitesimal” numbers, which were both nonzero but small enough to add to any real number without changing it noticeably. Although intuitively clear, infinitesi- mals were ultimately rejected as mathematically unsound, and were replaced with the common -δ method of computing limits and derivatives. However, in 1960 Abraham Robinson developed nonstandard analysis, in which the reals are rigor- ously extended to include infinitesimal numbers and infinite numbers; this new extended field is called the field of hyperreal numbers. The goal was to create a system of analysis that was more intuitively appealing than standard analysis but without losing any of the rigor of standard analysis. In this paper, we will explore the construction and various uses of nonstandard analysis. In section 2 we will introduce the notion of an ultrafilter, which will allow us to do a typical ultrapower construction of the hyperreal numbers. -
An Very Brief Overview of Surreal Numbers for Gandalf MM 2014
An very brief overview of Surreal Numbers for Gandalf MM 2014 Steven Charlton 1 History and Introduction Surreal numbers were created by John Horton Conway (of Game of Life fame), as a greatly simplified construction of an earlier object (Alling’s ordered field associated to the class of all ordinals, as constructed via modified Hahn series). The name surreal numbers was coined by Donald Knuth (of TEX and the Art of Computer Programming fame) in his novel ‘Surreal Numbers’ [2], where the idea was first presented. Surreal numbers form an ordered Field (Field with a capital F since surreal numbers aren’t a set but a class), and are in some sense the largest possible ordered Field. All other ordered fields, rationals, reals, rational functions, Levi-Civita field, Laurent series, superreals, hyperreals, . , can be found as subfields of the surreals. The definition/construction of surreal numbers leads to a system where we can talk about and deal with infinite and infinitesimal numbers as naturally and consistently as any ‘ordinary’ number. In fact it let’s can deal with even more ‘wonderful’ expressions 1 √ 1 ∞ − 1, ∞, ∞, ,... 2 ∞ in exactly the same way1. One large area where surreal numbers (or a slight generalisation of them) finds application is in the study and analysis of combinatorial games, and game theory. Conway discusses this in detail in his book ‘On Numbers and Games’ [1]. 2 Basic Definitions All surreal numbers are constructed iteratively out of two basic definitions. This is an wonderful illustration on how a huge amount of structure can arise from very simple origins. -
Cauchy, Infinitesimals and Ghosts of Departed Quantifiers 3
CAUCHY, INFINITESIMALS AND GHOSTS OF DEPARTED QUANTIFIERS JACQUES BAIR, PIOTR BLASZCZYK, ROBERT ELY, VALERIE´ HENRY, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TARAS KUDRYK, SEMEN S. KUTATELADZE, THOMAS MCGAFFEY, THOMAS MORMANN, DAVID M. SCHAPS, AND DAVID SHERRY Abstract. Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction be- tween assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robin- son’s framework, while Leibniz’s law of homogeneity with the im- plied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide paral- lel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Eu- ler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infi- nite exponent. Such procedures have immediate hyperfinite ana- logues in Robinson’s framework, while in a Weierstrassian frame- work they can only be reinterpreted by means of paraphrases de- parting significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of de- arXiv:1712.00226v1 [math.HO] 1 Dec 2017 parted quantifiers in his work. -
Some Mathematical and Physical Remarks on Surreal Numbers
Journal of Modern Physics, 2016, 7, 2164-2176 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Some Mathematical and Physical Remarks on Surreal Numbers Juan Antonio Nieto Facultad de Ciencias Fsico-Matemáticas de la Universidad Autónoma de Sinaloa, Culiacán, México How to cite this paper: Nieto, J.A. (2016) Abstract Some Mathematical and Physical Remarks on Surreal Numbers. Journal of Modern We make a number of observations on Conway surreal number theory which may be Physics, 7, 2164-2176. useful, for further developments, in both mathematics and theoretical physics. In http://dx.doi.org/10.4236/jmp.2016.715188 particular, we argue that the concepts of surreal numbers and matroids can be linked. Received: September 23, 2016 Moreover, we established a relation between the Gonshor approach on surreal num- Accepted: November 21, 2016 bers and tensors. We also comment about the possibility to connect surreal numbers Published: November 24, 2016 with supersymmetry. In addition, we comment about possible relation between sur- real numbers and fractal theory. Finally, we argue that the surreal structure may pro- Copyright © 2016 by author and Scientific Research Publishing Inc. vide a different mathematical tool in the understanding of singularities in both high This work is licensed under the Creative energy physics and gravitation. Commons Attribution International License (CC BY 4.0). Keywords http://creativecommons.org/licenses/by/4.0/ Open Access Surreal Numbers, Supersymmetry, Cosmology 1. Introduction Surreal numbers are a fascinating subject in mathematics. Such numbers were invented, or discovered, by the mathematician John Horton Conway in the 70’s [1] [2]. -
Do We Need Number Theory? II
Do we need Number Theory? II Václav Snášel What is a number? MCDXIX |||||||||||||||||||||| 664554 0xABCD 01717 010101010111011100001 푖푖 1 1 1 1 1 + + + + + ⋯ 1! 2! 3! 4! VŠB-TUO, Ostrava 2014 2 References • Z. I. Borevich, I. R. Shafarevich, Number theory, Academic Press, 1986 • John Vince, Quaternions for Computer Graphics, Springer 2011 • John C. Baez, The Octonions, Bulletin of the American Mathematical Society 2002, 39 (2): 145–205 • C.C. Chang and H.J. Keisler, Model Theory, North-Holland, 1990 • Mathématiques & Arts, Catalogue 2013, Exposants ESMA • А.Т.Фоменко, Математика и Миф Сквозь Призму Геометрии, http://dfgm.math.msu.su/files/fomenko/myth-sec6.php VŠB-TUO, Ostrava 2014 3 Images VŠB-TUO, Ostrava 2014 4 Number construction VŠB-TUO, Ostrava 2014 5 Algebraic number An algebraic number field is a finite extension of ℚ; an algebraic number is an element of an algebraic number field. Ring of integers. Let 퐾 be an algebraic number field. Because 퐾 is of finite degree over ℚ, every element α of 퐾 is a root of monic polynomial 푛 푛−1 푓 푥 = 푥 + 푎1푥 + … + 푎1 푎푖 ∈ ℚ If α is root of polynomial with integer coefficient, then α is called an algebraic integer of 퐾. VŠB-TUO, Ostrava 2014 6 Algebraic number Consider more generally an integral domain 퐴. An element a ∈ 퐴 is said to be a unit if it has an inverse in 퐴; we write 퐴∗ for the multiplicative group of units in 퐴. An element 푝 of an integral domain 퐴 is said to be irreducible if it is neither zero nor a unit, and can’t be written as a product of two nonunits. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Quantitative Tauberian Theorems
Quantitative Tauberian theorems omas Powell University of Bath Mathematical Logic: Proof Theory, Constructive Mathematics MFO (delivered online) 11 November 2020 ese slides are available at https://t-powell.github.io/talks Introduction Abel’s theorem Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever 1 X ai = s i=0 it follows that F(x) ! s as x % 1: is is a classical result in elementary analysis called Abel’s theorem (N.b. it also holds in the complex setting). You can use it to e.g. prove that 1 X (−1)i = log(2): i + 1 i=0 Does the converse of Abel’s theorem hold? NO. For a counterexample, define F :(−1; 1) ! R by 1 1 X F(x) = = (−1)ixi 1 + x i=0 en 1 F(x) ! as x % 1 2 but 1 X (−1)i does not converge i=0 Tauber’s theorem fixes this Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever F(x) ! s as x % 1 AND jnanj ! 0 it follows that 1 X ai = s i=0 is is Tauber’s theorem, proven in 1897 by Austrian mathematician Alfred Tauber (1866 - 1942). Tauberian theorems e basic structure of Tauber’s theorem is: 1 X i Let F(x) = aix i=0 en if we know (A) Something about the behaviour of F(x) as x % 1 (B) Something about the growth of fang as n ! 1 en we can conclude P1 (C) Something about the convergence of i=0 ai. -
Do Simple Infinitesimal Parts Solve Zeno's Paradox of Measure?
Do Simple Infinitesimal Parts Solve Zeno's Paradox of Measure?∗ Lu Chen (Forthcoming in Synthese) Abstract In this paper, I develop an original view of the structure of space|called infinitesimal atomism|as a reply to Zeno's paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson's (1966) nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite region is the sum of the sizes of its infinitesimal parts. Although this view is a coherent approach to Zeno's paradox and is preferable to Skyrms's (1983) infinitesimal approach, it faces both the main problem for the standard view (the problem of unmeasurable regions) and the main problem for finite atomism (Weyl's tile argument), leaving it with no clear advantage over these familiar alternatives. Keywords: continuum; Zeno's paradox of measure; infinitesimals; unmeasurable ∗Special thanks to Jeffrey Russell and Phillip Bricker for their extensive feedback on early drafts of the paper. Many thanks to the referees of Synthese for their very helpful comments. I'd also like to thank the participants of Umass dissertation seminar for their useful feedback on my first draft. 1 regions; Weyl's tile argument 1 Zeno's Paradox of Measure A continuum, such as the region of space you occupy, is commonly taken to be indefinitely divisible. But this view runs into Zeno's famous paradox of measure. -
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V. -
The Littlewood Tauberian Theorem
Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 1 The Littlewood Tauberian theorem 1.1 Introduction In 1897, the Austrian mathematician Alfred Tauber published a short article on the convergence of numerical series [173], which can be summarised as follows. ∞ = Let an be a convergent series of complex numbers, with n=0 an .A theorem of Abel [1] states that ∞ n f (x) = an x → as x 1. (1.1) n=0 A theorem of Kronecker [116] states that n 1 → . n kak 0 (1.2) k=1 The converse of these two theorems is false: neither of the two conditions (1) nor (2) is sufficient to imply the convergence of the series an. However, if both conditions are satisfied simultaneously, the series an converges, giving the following theorem. 1.1.1 Theorem A necessary and sufficient condition for an to converge (with sum ) is that: ∞ (1) f (x) = a xn → as x 1, n=0 n 1 n → (2) n k=1 kak 0. The proof of Theorem 1.1.1 follows that of the following special case. 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-05945-0 - Twelve Landmarks of Twentieth-Century Analysis D. Choimet and H. Queffélec Excerpt More information 2 The Littlewood Tauberian theorem ( ) → → ∞ = 1.1.2 Theorem f x as x 1, and nan 0 implies n=0 an . A few remarks on this article: the theorem of Abel cited above gave rise to Abelian theorems [113], that is, theorems of the form If an is a convergent complex series with sum , and (bn,x ) an infinite rectangular matrix indexed by N × X, where X is a set with an associated point at infinity satisfying bn,x −→ 1 forevery n ∈ N, x→∞ then ∞ f (x) = anbn,x n=0 is defined for x ∈ X and f (x) → as x →∞. -
An Application of Tauber Theory: Proving the Prime Number Theorem
faculty of science mathematics and applied and engineering mathematics An application of Tauber theory: proving the Prime Number Theorem Bachelor’s Project Mathematics February 2018 Student: J. Koolstra Supervisors: Prof.dr. J. Top and Dr. A. Sterk Abstract We look at the origins of Tauber theory, and apply it to prove the prime number theorem (PNT). Specifically, we prove a weak version of the Wiener-Ikehara Tauberian theorem due to Newman. Its appli- cation requires us to establish some properties of the Riemann zeta function. Most notably with regard to its meromorphic continuation, and the distribution of its zeros. 2 Contents Introduction 5 Discrete Tauber theory 7 Summability methods . .7 The theorems of Tauber and Hardy-Littlewood . 13 The Wiener-Ikehara theorem 19 Some complex analytic technicalities . 19 A proof of the Wiener-Ikehara theorem . 19 The Riemann zeta function 28 The meromorphic continuation of the zeta function . 28 The zeta function is nonzero on the line Re z =1.......... 32 The prime number theorem 37 The linear bound on the Tchebychef -function . 37 Equivalent formulations of PNT . 39 Appendix A: the prime polynomial theorem 42 References 48 3 Even before I had begun my more detailed investigations into higher arithmetic, one of my first projects was to turn my attention to the decreasing frequency of primes, to which end I counted primes in several chiliads. I soon recognized that behind all of its fluctuations, this frequency is on average inversely proportional to the logarithm. Gauss to Encke, 1849 It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. -
Cantor and Continuity
Cantor and Continuity Akihiro Kanamori May 1, 2018 Georg Cantor (1845-1919), with his seminal work on sets and number, brought forth a new field of inquiry, set theory, and ushered in a way of proceeding in mathematics, one at base infinitary, topological, and combinatorial. While this was the thrust, his work at the beginning was embedded in issues and concerns of real analysis and contributed fundamentally to its 19th Century rigorization, a development turning on limits and continuity. And a continuing engagement with limits and continuity would be very much part of Cantor's mathematical journey, even as dramatically new conceptualizations emerged. Evolutionary accounts of Cantor's work mostly underscore his progressive ascent through set- theoretic constructs to transfinite number, this as the storied beginnings of set theory. In this article, we consider Cantor's work with a steady focus on con- tinuity, putting it first into the context of rigorization and then pursuing the increasingly set-theoretic constructs leading to its further elucidations. Beyond providing a narrative through the historical record about Cantor's progress, we will bring out three aspectual motifs bearing on the history and na- ture of mathematics. First, with Cantor the first mathematician to be engaged with limits and continuity through progressive activity over many years, one can see how incipiently metaphysical conceptualizations can become systemati- cally transmuted through mathematical formulations and results so that one can chart progress on the understanding of concepts. Second, with counterweight put on Cantor's early career, one can see the drive of mathematical necessity pressing through Cantor's work toward extensional mathematics, the increasing objectification of concepts compelled, and compelled only by, his mathematical investigation of aspects of continuity and culminating in the transfinite numbers and set theory.