DOCSLIB.ORG

Which Numbers Are Real? Are Numbers Which / MAA AMS

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

AMS / MAA CLASSROOM RESOURCE MATERIALS VOL AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 42 42 Which Numbers Are Real? Which Numbers are Real? MICHAEL HENLE Everyone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry THE SURREAL NUMBERS through the Calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics. Alternative real numbers include many different kinds of numbers, for example multidimen- THE HYPERREAL NUMBERS sional numbers (the complex numbers, the quaternions and others), infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). Each system has a well-developed theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multi-dimen- sional geometry, and formal logic. They are all active areas of current mathemat- THE CONSTRUCTIVE REALS ical research and each has unique features, in particular, characteristic methods of proof and implications for the philosophy of mathematics, both highlighted in this book. Alternative real number systems illuminate the central, unifying Michael Henle role of the real numbers and include some exciting and eccentric parts of math- ematics. Which Numbers Are Real? Will be of interest to anyone with an interest in THE COMPLEX NUMBERS numbers, but specifically to upper-level undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers. THE QUATERNIONS AMS/ MAA PRESS 4-Color Process 230 pages on 50lb stock • trim size: 6"x9" • Spine 7/16" i i “master” — 2012/6/18 — 10:53 — page i — #1 i i 10.1090/clrm/042 Which Numbers are Real? i i i i i i “master” — 2012/6/18 — 10:53 — page ii — #2 i i c 2012 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2012937493 Print edition ISBN 978-0-88385-777-9 Electronic edition ISBN 978-1-61444-107-6 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “master” — 2012/6/18 — 10:53 — page iii — #3 i i Which Numbers are Real? Michael Henle Oberlin College Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2012/6/18 — 10:53 — page iv — #4 i i Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Philip P. Mummert Barbara E. Reynolds Susan G. Staples Philip D. Straffin Cynthia J Woodburn Holly S. Zullo i i i i i i “master” — 2012/6/18 — 10:53 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary class- room material for students—laboratory exercises, projects, historical in- formation, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer, John Maceli, and Eric Robinson The Calculus Collection: A Resource for AP and Beyond, edited by Caren L. Diefenderfer and Roger B. Nelsen Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Mikl´os Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research, by Hongwei Chen Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry From Africa: Mathematical and EducationalExplorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for UnderstandingMathematics, Claudi Alsina and Roger B. Nelsen Mathematics Galore!: The First Five Years of the St. Marks Institute of Mathematics, James Tanton i i i i i i “master” — 2012/6/18 — 10:53 — page vi — #6 i i Ordinary Differential Equations: A Brief Eclectic Tour, David A. S´anchez Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen Rediscovering Mathematics: You Do the Math, Shai Simonson She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probabil- ity and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “master” — 2012/6/18 — 10:53 — page vii — #7 i i Introduction Alternative reality The real numbers are fundamental. Although mostly taken for granted, they are what make possible all of mathematics from high school algebra and Euclidean geometry through the calculus and beyond, and also serve as the basis for measurement in science, industry, and ordinary life. In this book we study alternative systems of numbers: systems that generalize and ex- tend the reals yet stay close to the fundamental properties that make the reals central to so much mathematics. By an alternative number system we mean a set of objects that can be combined using two operations, addition and multiplication, and that share some significant algebraic and geometric properties with the real numbers. Exactly what these properties are is made clear in Chapter One. We are not concerned with numeration, however. A numeration system is a means of giving names to numbers, for example, the decimal system for writing real numbers. We go beyond numeration to describe number systems that in- clude numbers differentfrom ordinary numbers includingmulti-dimensional numbers, infinitely small and infinitely large numbers, and numbers that represent positions in games. Although we present some eccentric and relatively unexplored parts of mathematics, each system that we study has a well-developed theory. Each system has applications to other areas of mathematics and science, in par- ticular to physics, the theory of games, multi-dimensional geometry, formal logic, and the philosophy of mathematics. Most of these number systems are active areas of current mathematical research and several were discov- vii i i i i i i “master” — 2012/6/18 — 10:53 — page viii — #8 i i viii Introduction ered relatively recently. As a group, they illuminate the central, unifying role of the reals in mathematics. Design of this book This book is designed to encourage readers to participate in the mathemat- ical development themselves. The proofs of many results are either con- tained in problems or depend on results proved in problems. The problems should be read at least, if not worked out. With twoexceptions the chapters are independent and can be read in any order. The exceptions are that the first two chapters contain essential back- ground for the rest of the book and that Chapter Four depends somewhat on the proceeding chapter. Use of this book This book presents material that, in addition to being of general interest to mathematics students, is appropriate for an upper level course for under- graduates that can serve as introductionto or sequel to a course in advanced calculus. Alternatively, it can take the place of a course in the foundations of the real number system, or be given as an upper level seminar emphasiz- ing different methods of proof. Prerequisites are standard sophomore level courses: discrete mathematics, multivariable calculus, and linear algebra. A course in advanced calculus or foundationsof analysis would also be useful. The goal The goal is to present some interesting, even exotic, mathematics. I hope to convey a sense of the immense freedom available in mathematics, where even in a mundane and well-established area such as the real numbers, al- ternatives are always possible. Acknowledgements This book owes a tremendous debt to previous expositors in this area to whose work this book is closely tied, in particular, to the work of Elwyn Berlekamp, Errett Bishop, John H. Conway, Richard Guy, James Henle, and Eugene Kleinberg. i i i i i i “master” — 2012/6/18 — 10:53 — page ix — #9 i i Contents Introduction vii I THE REALS 1 1 Axiomsfor
Recommended publications
  • An Introduction to Nonstandard Analysis 11

    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

    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 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

    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].
  • An Introduction to Conway's Games and Numbers

    An Introduction to Conway's Games and Numbers

    AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS DIERK SCHLEICHER AND MICHAEL STOLL 1. Combinatorial Game Theory Combinatorial Game Theory is a fascinating and rich theory, based on a simple and intuitive recursive definition of games, which yields a very rich algebraic struc- ture: games can be added and subtracted in a very natural way, forming an abelian GROUP (§ 2). There is a distinguished sub-GROUP of games called numbers which can also be multiplied and which form a FIELD (§ 3): this field contains both the real numbers (§ 3.2) and the ordinal numbers (§ 4) (in fact, Conway’s definition gen- eralizes both Dedekind sections and von Neumann’s definition of ordinal numbers). All Conway numbers can be interpreted as games which can actually be played in a natural way; in some sense, if a game is identified as a number, then it is under- stood well enough so that it would be boring to actually play it (§ 5). Conway’s theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go [Go]. There is a beautiful microcosmos of numbers and games which are infinitesimally close to zero (§ 6), and the theory contains the classical and complete Sprague-Grundy theory on impartial games (§ 7). The theory was founded by John H. Conway in the 1970’s. Classical references are the wonderful books On Numbers and Games [ONAG] by Conway, and Win- ning Ways by Berlekamp, Conway and Guy [WW]; they have recently appeared in their second editions.
  • Do We Need Number Theory? II

    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

    0.999… = 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 .
  • Do Simple Infinitesimal Parts Solve Zeno's Paradox of Measure?

    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

    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.
  • Lecture 25: Ultraproducts

    Lecture 25: Ultraproducts

    LECTURE 25: ULTRAPRODUCTS CALEB STANFORD First, recall that given a collection of sets and an ultrafilter on the index set, we formed an ultraproduct of those sets. It is important to think of the ultraproduct as a set-theoretic construction rather than a model- theoretic construction, in the sense that it is a product of sets rather than a product of structures. I.e., if Xi Q are sets for i = 1; 2; 3;:::, then Xi=U is another set. The set we use does not depend on what constant, function, and relation symbols may exist and have interpretations in Xi. (There are of course profound model-theoretic consequences of this, but the underlying construction is a way of turning a collection of sets into a new set, and doesn't make use of any notions from model theory!) We are interested in the particular case where the index set is N and where there is a set X such that Q Xi = X for all i. Then Xi=U is written XN=U, and is called the ultrapower of X by U. From now on, we will consider the ultrafilter to be a fixed nonprincipal ultrafilter, and will just consider the ultrapower of X to be the ultrapower by this fixed ultrafilter. It doesn't matter which one we pick, in the sense that none of our results will require anything from U beyond its nonprincipality. The ultrapower has two important properties. The first of these is the Transfer Principle. The second is @0-saturation. 1. The Transfer Principle Let L be a language, X a set, and XL an L-structure on X.
  • SURREAL TIME and ULTRATASKS HAIDAR AL-DHALIMY Department of Philosophy, University of Minnesota and CHARLES J

    SURREAL TIME and ULTRATASKS HAIDAR AL-DHALIMY Department of Philosophy, University of Minnesota and CHARLES J

    THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 12 SURREAL TIME AND ULTRATASKS HAIDAR AL-DHALIMY Department of Philosophy, University of Minnesota and CHARLES J. GEYER School of Statistics, University of Minnesota Abstract. This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. §1. Introduction. To perform a hypertask is to complete an uncountable sequence of tasks in a finite length of time. Each task is taken to require a nonzero interval of time, with the intervals being pairwise disjoint. Clark & Read (1984) claim to “show that no hypertask can be performed” (p. 387). What they in fact show is the impossibility of a situation in which both (i) a hypertask is performed and (ii) time has the structure of the real number system as conceived of in mainstream mathematics—or, as we will say, time is R-like. However, they make the argument overly complicated.
  • Exploring Euler's Foundations of Differential Calculus in Isabelle

    Exploring Euler's Foundations of Differential Calculus in Isabelle

    Exploring Euler’s Foundations of Differential Calculus in Isabelle/HOL using Nonstandard Analysis Jessika Rockel I V N E R U S E I T H Y T O H F G E R D I N B U Master of Science Computer Science School of Informatics University of Edinburgh 2019 Abstract When Euler wrote his ‘Foundations of Differential Calculus’ [5], he did so without a concept of limits or a fixed notion of what constitutes a proof. Yet many of his results still hold up today, and he is often revered for his skillful handling of these matters despite the lack of a rigorous formal framework. Nowadays we not only have a stricter notion of proofs but we also have computer tools that can assist in formal proof development: Interactive theorem provers help users construct formal proofs interactively by verifying individual proof steps and pro- viding automation tools to help find the right rules to prove a given step. In this project we examine the section of Euler’s ‘Foundations of Differential Cal- culus’ dealing with the differentiation of logarithms [5, pp. 100-104]. We retrace his arguments in the interactive theorem prover Isabelle to verify his lines of argument and his conclusions and try to gain some insight into how he came up with them. We are mostly able to follow his general line of reasoning, and we identify a num- ber of hidden assumptions and skipped steps in his proofs. In one case where we cannot reproduce his proof directly we can still validate his conclusions, providing a proof that only uses methods that were available to Euler at the time.