DOCSLIB.ORG

Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets by Rebecca Stettin A capstone project submitted in partial fulfillment of graduating from the Academic Honors Program at Ashland University May 2017 Faculty Mentor: Dr. Darren D. Wick, Professor of Mathematics Additional Reader: Dr. Gordon Swain, Professor of Mathematics Abstract Georg Cantor was made famous by introducing the Cantor set in his works of mathemat- ics. This project focuses on different Cantor sets and their properties. The ternary Cantor set is the most well known of the Cantor sets, and can be best described by its construction. This set starts with the closed interval zero to one, and is constructed in iterations. The first iteration requires removing the middle third of this interval. The second iteration will remove the middle third of each of these two remaining intervals. These iterations continue in this fashion infinitely. Finally, the ternary Cantor set is described as the intersection of all of these intervals. This set is particularly interesting due to its unique properties being uncountable, closed, length of zero, and more. A more general Cantor set is created by tak- ing the intersection of iterations that remove any middle portion during each iteration. This project explores the ternary Cantor set, as well as variations in Cantor sets such as looking at different middle portions removed to create the sets. The project focuses on attempting to generalize the properties of these Cantor sets. i Contents Page 1 The Ternary Cantor Set 1 1 2 The n -ary Cantor Set 9 n−1 3 The n -ary Cantor Set 24 4 Conclusion 35 Bibliography 40 Biography 41 ii Chapter 1 The Ternary Cantor Set Georg Cantor, born in 1845, was best known for his discovery of the Cantor set. After writing a thesis in number theory, Cantor became interested in topology and wrote a series of papers on point set topology. He coined the term everywhere dense, which is currently used today. In this same series of papers, Cantor later introduced the idea of perfect sets. He made the ternary Cantor set famous in this same paper when attempting to prove that it was possible to have a set that is closed and nowhere dense. He noted that this was a perfect, infinite set that is nowhere dense in any interval, regardless of the size. (Fleron [3]) The Cantor set is created by starting with the closed interval [0,1]. The next step is to remove a middle portion of the interval. This initial removal will then leave two intervals, namely 1 2 [0, 3 ] and [ 3 , 1]. This process is continued an infinite number of times, with the final set of numbers, after the infinite iteration, being the Cantor set itself. This set is the intersection 1 th of all iterations, denoted as T = \i=1 Ti, where Ti represents the i iteration. There are 1 different proportions that can be removed with this set, the most well known being 3 . This set, referred to as the ternary Cantor set, can also be referred to as the classic set. This particular version of the Cantor set was the one Georg Cantor used to make his arguments about the nature of certain sets. (Aczel [1]) We will describe the construction of the ternary Cantor set, T , in a similar manner as it is 1 described in Thomson [7]. Begin with the closed interval [0; 1] and remove a dense open set, G. The remaining set, T = [0; 1] nG will also be closed and nowhere dense in [0; 1]. Based on our construction of G, T will have no isolated points. It is easiest to understand the set 1 2 G if we construct it in stages. Let G1 = ( 3 ; 3 ) and let T1 be what is remaining in [0; 1] after 1 2 removing G1. Thus T1 = [0, 3 ] [ [ 3 , 1] is what remains when the middle third of the interval [0,1] is removed. This is referred to as the first iteration of the ternary Cantor set. We repeat 1 2 7 8 this construction for each of the two intervals of T1. Let G2 = ( 9 ; 9 ) [ ( 9 ; 9 ). These intervals 1 2 1 2 7 8 are the middle third of the previous 2 intervals. Then T2 = [0; 9 ] [ [ 9 ; 3 ] [ [ 3 ; 9 ] [ [ 9 ; 1]. This completes the second iteration. We continue inductively, and ultimately take the intersection of each iteration of Ti to create the ternary Cantor set with the following properties: For each i 2 N i 1. Ti is a union of 2 pairwise disjoint closed intervals. 2. Each component of the set of open intervals removed is the "middle third" of some component of Ti. 1 3. The length of each component of Ti is 3i . Some proofs of the properties of the ternary Cantor set will be explored in later chapters in regards to more generalized Cantor sets. The following is a list of characteristics of the ternary Cantor set with some proofs to follow: 1. The Cantor set is nonempty. 2. The Cantor set is closed. 3. The Cantor set has length 0. 4. The Cantor set contains all numbers that can be written in base-3 without 1's. 5. The Cantor set is uncountable. 6. The Cantor set is nowhere dense. 2 7. The Cantor set has no isolated points. 8. The Cantor set is a perfect set. Theorem 1. The ternary Cantor set is nonempty. Proof. Using similar methods as Nelson [5], we must show that the ternary Cantor set contains at least one element. We can see that during each iteration, the endpoints remain 1 2 in the set. For example, during T1, everything between ( 3 , 3 ) is removed. Note that it is everything between the two endpoints, which implies the two endpoints remain. Once 1 2 removed, T1 is [0, 3 ] [ [ 3 , 1]. As seen in here, the intervals are closed, thus including the endpoints. This trend continues throughout each iteration. For T2, the middle thirds of 1 those two previously mentioned intervals are removed. To clarify, everything between ( 9 , 2 7 8 1 2 1 2 7 8 9 ) as well as ( 9 , 9 ) is removed. This leaves [0, 9 ] [ [ 9 , 3 ] [ [ 3 , 9 ] [ [ 9 , 1], which is once 1 2 again a union of closed intervals, which include endpoints. Notice that 0, 3 , 3 , and 1 are again included in this iteration. These iterations continue infinitely. Since each iteration of T contains the endpoints of each iteration before it, it becomes clear that the endpoints of each iteration are included. Since the previous endpoints are consistently included, there will clearly be elements in the intersection of each iteration, proving the ternary Cantor set is nonempty. Theorem 2. The ternary Cantor set is closed. Proof. As shown earlier when looking at the length of T, it is an intersection of closed sets. 1 2 For example, T1 is [0, 3 ] [ [ 3 , 1]. This trend continues with each Ti also consisting of a finite union of closed intervals. Each Ti is closed since it is a finite union of closed intervals. 1 Since T = \i=1Ti it is a countable intersection of closed sets and is therefore closed. Theorem 3. The ternary Cantor set has length of 0. Proof. This proof is also based off of similar methods used in Nelson [5]. This proof can be demonstrated by noting everything that is taken out of the starting interval has the same 3 length as the initial interval. Looking at what is taken out during each iteration, a sequence 1 emerges. During the first iteration, an interval of length 3 is removed from the set. During 1 2 the second iteration, there are 2 different intervals, namely [0, 3 ] and [ 3 ,1], to remove the 1 1 1 middle third from. In this case, however, the length of what is removed is 3 of 3 , or 9 . This 1 th must occur twice because there are 2 intervals that ( 9 ) is being removed from. Therefore, 2 2 a length of 32 = 9 will be removed during the second iteration of T. This trend continues 1 2 4 8 throughout each iteration resulting in the length removed at each iteration as 3 , 9 , 27 , 81 ,··· The total length removed can also be represented as 1 X 2i 3i+1 i=0 1 1 This sum can be determined through the sum as a geometric series, or 3 2 = 1. This 1− 3 means that everything removed has length 1. As mentioned previously, the starting interval, T0, has length 1 as well. A simple subtraction of the removed portion from the initial length will give the final length of T as 0. Therefore, T has length 0. The following theorem will be explored through examples, but will not be proven. A similar proof can be found in Chapter 3. Theorem 4. The ternary Cantor set only contains numbers that have a base-3 representation excluding the digit 1. A brief review of base-3 representation is necessary to proceed with this example. Base-3, or ternary, representation consists of 3 digits, 0, 1, and 2. A number of the form (.a1a2a3··· )3 n can be thought of in terms of sums of fractions with numerators an and denominators 3 . a1 a2 a3 That is, (.a1a2a3··· )3 = 3 + 32 + 33 + ··· where an must be either 0, 1, or 2. The claim is every number that is in the ternary Cantor set can be written in base-3 representation with only 2's and 0's.
Load more

Recommended publications
  • On the Structures of Generating Iterated Function Systems of Cantor Sets
    ON THE STRUCTURES OF GENERATING ITERATED FUNCTION SYSTEMS OF CANTOR SETS DE-JUN FENG AND YANG WANG Abstract. A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under the open set condition (OSC). If dimH F < 1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study. 1. Introduction N d In this paper, a family of contractive affine maps Φ = fφjgj=1 in R is called an iterated function system (IFS). According to Hutchinson [12], there is a unique non-empty compact d SN F = FΦ ⊂ R , which is called the attractor of Φ, such that F = j=1 φj(F ). Furthermore, FΦ is called a self-similar set if Φ consists of similitudes. It is well known that the standard middle-third Cantor set C is the attractor of the iterated function system (IFS) fφ0; φ1g where 1 1 2 (1.1) φ (x) = x; φ (x) = x + : 0 3 1 3 3 A natural question is: Is it possible to express C as the attractor of another IFS? Surprisingly, the general question whether the attractor of an IFS can be expressed as the attractor of another IFS, which seems a rather fundamental question in fractal geometry, has 1991 Mathematics Subject Classification.
    [Show full text]
  • Georg Cantor English Version
    GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle.
    [Show full text]
  • Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions
    Journal of Applied Mathematics and Computation, 2020, 4(4), 123-129 https://www.hillpublisher.com/journals/JAMC/ ISSN Online: 2576-0653 ISSN Print: 2576-0645 Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions Srijanani Anurag Prasad Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, India. How to cite this paper: Srijanani Anurag Prasad. (2020) Riemann-Liouville Frac- Abstract tional Calculus of Blancmange Curve and Riemann-Liouville fractional calculus of Blancmange Curve and Cantor Func- Cantor Functions. Journal of Applied Ma- thematics and Computation, 4(4), 123-129. tions are studied in this paper. In this paper, Blancmange Curve and Cantor func- DOI: 10.26855/jamc.2020.12.003 tion defined on the interval is shown to be Fractal Interpolation Functions with appropriate interpolation points and parameters. Then, using the properties of Received: September 15, 2020 Fractal Interpolation Function, the Riemann-Liouville fractional integral of Accepted: October 10, 2020 Published: October 22, 2020 Blancmange Curve and Cantor function are described to be Fractal Interpolation Function passing through a different set of points. Finally, using the conditions for *Corresponding author: Srijanani the fractional derivative of order ν of a FIF, it is shown that the fractional deriva- Anurag Prasad, Department of Mathe- tive of Blancmange Curve and Cantor function is not a FIF for any value of ν. matics, Indian Institute of Technology Tirupati, India. Email: [email protected] Keywords Fractal, Interpolation, Iterated Function System, fractional integral, fractional de- rivative, Blancmange Curve, Cantor function 1. Introduction Fractal geometry is a subject in which irregular and complex functions and structures are researched.
    [Show full text]
  • Descriptive Set Theory
    Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 E®ective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic Sets 34 5 Coanalytic Sets 43 6 Determinacy 54 7 Hyperarithmetic Sets 62 II Borel Equivalence Relations 73 1 8 ¦1-Equivalence Relations 73 9 Tame Borel Equivalence Relations 82 10 Countable Borel Equivalence Relations 87 11 Hyper¯nite Equivalence Relations 92 1 These are informal notes for a course in Descriptive Set Theory given at the University of Illinois at Chicago in Fall 2002. While I hope to give a fairly broad survey of the subject we will be concentrating on problems about group actions, particularly those motivated by Vaught's conjecture. Kechris' Classical Descriptive Set Theory is the main reference for these notes. Notation: If A is a set, A<! is the set of all ¯nite sequences from A. Suppose <! σ = (a0; : : : ; am) 2 A and b 2 A. Then σ b is the sequence (a0; : : : ; am; b). We let ; denote the empty sequence. If σ 2 A<!, then jσj is the length of σ. If f : N ! A, then fjn is the sequence (f(0); : : :b; f(n ¡ 1)). If X is any set, P(X), the power set of X is the set of all subsets X. If X is a metric space, x 2 X and ² > 0, then B²(x) = fy 2 X : d(x; y) < ²g is the open ball of radius ² around x. Part I Classical Descriptive Set Theory 1 Polish Spaces De¯nition 1.1 Let X be a topological space.
    [Show full text]
  • Fractal Geometry and Applications in Forest Science
    ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals ....................................................................................
    [Show full text]
  • Fractals Lindenmayer Systems
    FRACTALS LINDENMAYER SYSTEMS November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin RECAP HIDDEN MARKOV MODELS What Letter Is Written Here? What Letter Is Written Here? What Letter Is Written Here? The Idea Behind Hidden Markov Models First letter: Maybe „a“, maybe „q“ Second letter: Maybe „r“ or „v“ or „u“ Take the most probable combination as a guess! Hidden Markov Models Sometimes, you don‘t see the states, but only a mapping of the states. A main task is then to derive, from the visible mapped sequence of states, the actual underlying sequence of „hidden“ states. HMM: A Fundamental Question What you see are the observables. But what are the actual states behind the observables? What is the most probable sequence of states leading to a given sequence of observations? The Viterbi-Algorithm We are looking for indices M1,M2,...MT, such that P(qM1,...qMT) = Pmax,T is maximal. 1. Initialization ()ib 1 i i k1 1(i ) 0 2. Recursion (1 t T-1) t1(j ) max( t ( i ) a i j ) b j k i t1 t1(j ) i : t ( i ) a i j max. 3. Termination Pimax,TT max( ( )) qmax,T q i: T ( i ) max. 4. Backtracking MMt t11() t Efficiency of the Viterbi Algorithm • The brute force approach takes O(TNT) steps. This is even for N = 2 and T = 100 difficult to do. • The Viterbi – algorithm in contrast takes only O(TN2) which is easy to do with todays computational means. Applications of HMM • Analysis of handwriting. • Speech analysis. • Construction of models for prediction.
    [Show full text]
  • Set Ideal Topological Spaces
    Set Ideal Topological Spaces W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2012 by Zip Publishing and the Authors Peer reviewers: Prof. Catalin Barbu, V. Alecsandri National College, Mathematics Department, Bacau, Romania. Prof. Valeri Kroumov, Okayama Univ. of Science, Japan. Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-193-3 EAN: 9781599731933 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET IDEALS IN RINGS 9 Chapter Three SET IDEAL TOPOLOGICAL SPACES 35 Chapter Four NEW CLASSES OF SET IDEAL TOPOLOGICAL SPACES AND APPLICATIONS 93 3 FURTHER READING 109 INDEX 111 ABOUT THE AUTHORS 114 4 PREFACE In this book the authors for the first time introduce a new type of topological spaces called the set ideal topological spaces using rings or semigroups, or used in the mutually exclusive sense. This type of topological spaces use the class of set ideals of a ring (semigroups). The rings or semigroups can be finite or infinite order. By this method we get complex modulo finite integer set ideal topological spaces using finite complex modulo integer rings or finite complex modulo integer semigroups. Also authors construct neutrosophic set ideal toplogical spaces of both finite and infinite order as well as complex neutrosophic set ideal topological spaces.
    [Show full text]
  • ON BERNSTEIN SETS Let Us Recall That a Subset of a Topological Space
    ON BERNSTEIN SETS JACEK CICHON´ ABSTRACT. In this note I show a construction of Bernstein subsets of the real line which gives much more information about the structure of the real line than the classical one. 1. BASIC DEFINITIONS Let us recall that a subset of a topological space is a perfect set if is closed set and contains no isolated points. Perfect subsets of real line R have cardinality continuum. In fact every perfect set contains a copy of a Cantor set. This can be proved rather easilly by constructing a binary tree of decreasing small closed sets. There are continnum many perfect sets. This follows from a more general result: there are continuum many Borel subsets of real. We denote by R the real line. We will work in the theory ZFC. Definition 1. A subset B of the real line R is a Bernstein set if for every perfect subset P of R we have (P \ B 6= ;) ^ (P n B 6= ;) : Bernstein sets are interesting object of study because they are Lebesgue nonmeasurable and they do not have the property of Baire. The classical construction of a Berstein set goes as follows: we fix an enumeration (Pα)α<c of all perfect sets, we define a transfinite sequences (pα), (qα) of pairwise different points such that fpα; qαg ⊆ Pα, we put B = fpα : α < cg and we show that B is a Bernstein set. This construction can be found in many classical books. The of this note is to give slightly different construction, which will generate simultanously a big family of Bermstein sets.We start with one simple but beautifull result: Lemma 1.
    [Show full text]
  • Lecture 8: the Real Line, Part II February 11, 2009
    Lecture 8: The Real Line, Part II February 11, 2009 Definition 6.13. a ∈ X is isolated in X iff there is an open interval I for which X ∩ I = {a}. Otherwise, a is a limit point. Remark. Another way to state this is that a is isolated if it is not a limit point of X. Definition 6.14. X is a perfect set iff X is closed and has no isolated points. Remark. This definition sounds nice and tidy, but there are some very strange perfect sets. For example, the Cantor set is perfect, despite being nowhere dense! Our goal will be to prove the Cantor-Bendixson theorem, i.e. the perfect set theorem for closed sets, that every closed uncountable set has a perfect subset. Lemma 6.15. If P is a perfect set and I is an open interval on R such that I ∩ P 6= ∅, then there exist disjoint closed intervals J0,J1 ⊂ I such that int[J0] ∩ P 6= ∅ and int[J1] ∩ P 6= ∅. Moreover, we can pick J0 and J1 such that their lengths are both less than any ² > 0. Proof. Since P has no isolated points, there must be at least two points a0, a1 ∈ I ∩ P . Then just pick J0 3 a0 and J1 3 a1 to be small enough. SDG Lemma 6.16. If P is a nonempty perfect set, then P ∼ R. Proof. We exhibit a one-to-one mapping G : 2ω → P . Note that 2ω can be viewed as the set of all infinite paths in a full, infinite binary tree with each edge labeled by 0 or 1.
    [Show full text]
  • Chapter 6 Introduction to Calculus
    RS - Ch 6 - Intro to Calculus Chapter 6 Introduction to Calculus 1 Archimedes of Syracuse (c. 287 BC – c. 212 BC ) Bhaskara II (1114 – 1185) 6.0 Calculus • Calculus is the mathematics of change. •Two major branches: Differential calculus and Integral calculus, which are related by the Fundamental Theorem of Calculus. • Differential calculus determines varying rates of change. It is applied to problems involving acceleration of moving objects (from a flywheel to the space shuttle), rates of growth and decay, optimal values, etc. • Integration is the "inverse" (or opposite) of differentiation. It measures accumulations over periods of change. Integration can find volumes and lengths of curves, measure forces and work, etc. Older branch: Archimedes (c. 287−212 BC) worked on it. • Applications in science, economics, finance, engineering, etc. 2 1 RS - Ch 6 - Intro to Calculus 6.0 Calculus: Early History • The foundations of calculus are generally attributed to Newton and Leibniz, though Bhaskara II is believed to have also laid the basis of it. The Western roots go back to Wallis, Fermat, Descartes and Barrow. • Q: How close can two numbers be without being the same number? Or, equivalent question, by considering the difference of two numbers: How small can a number be without being zero? • Fermat’s and Newton’s answer: The infinitessimal, a positive quantity, smaller than any non-zero real number. • With this concept differential calculus developed, by studying ratios in which both numerator and denominator go to zero simultaneously. 3 6.1 Comparative Statics Comparative statics: It is the study of different equilibrium states associated with different sets of values of parameters and exogenous variables.
    [Show full text]
  • History and Mythology of Set Theory
    Chapter udf History and Mythology of Set Theory This chapter includes the historical prelude from Tim Button's Open Set Theory text. set.1 Infinitesimals and Differentiation his:set:infinitesimals: Newton and Leibniz discovered the calculus (independently) at the end of the sec 17th century. A particularly important application of the calculus was differ- entiation. Roughly speaking, differentiation aims to give a notion of the \rate of change", or gradient, of a function at a point. Here is a vivid way to illustrate the idea. Consider the function f(x) = 2 x =4 + 1=2, depicted in black below: f(x) 5 4 3 2 1 x 1 2 3 4 1 Suppose we want to find the gradient of the function at c = 1=2. We start by drawing a triangle whose hypotenuse approximates the gradient at that point, perhaps the red triangle above. When β is the base length of our triangle, its height is f(1=2 + β) − f(1=2), so that the gradient of the hypotenuse is: f(1=2 + β) − f(1=2) : β So the gradient of our red triangle, with base length 3, is exactly 1. The hypotenuse of a smaller triangle, the blue triangle with base length 2, gives a better approximation; its gradient is 3=4. A yet smaller triangle, the green triangle with base length 1, gives a yet better approximation; with gradient 1=2. Ever-smaller triangles give us ever-better approximations. So we might say something like this: the hypotenuse of a triangle with an infinitesimal base length gives us the gradient at c = 1=2 itself.
    [Show full text]
  • Set Theory: Cantor
    Notes prepared by Stanley Burris March 13, 2001 Set Theory: Cantor As we have seen, the naive use of classes, in particular the connection be- tween concept and extension, led to contradiction. Frege mistakenly thought he had repaired the damage in an appendix to Vol. II. Whitehead & Russell limited the possible collection of formulas one could use by typing. Another, more popular solution would be introduced by Zermelo. But ¯rst let us say a few words about the achievements of Cantor. Georg Cantor (1845{1918) 1872 - On generalizing a theorem from the theory of trigonometric series. 1874 - On a property of the concept of all real algebraic numbers. 1879{1884 - On in¯nite linear manifolds of points. (6 papers) 1890 - On an elementary problem in the study of manifolds. 1895/1897 - Contributions to the foundation to the study of trans¯nite sets. We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather be- cause he initiated the study of in¯nite sets and numbers which have provided such fascinating material, and di±culties, for logicians. After all, a natural foundation for mathematics would need to talk about sets of real numbers, etc., and any reasonably expressive system should be able to cope with one- to-one correspondences and well-orderings. Cantor started his career by working in algebraic and analytic number theory. Indeed his PhD thesis, his Habilitation, and ¯ve papers between 1867 and 1880 were devoted to this area. At Halle, where he was employed after ¯nishing his studies, Heine persuaded him to look at the subject of trigonometric series, leading to eight papers in analysis.
    [Show full text]