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Infinity in Mathematics GLIMPSES OF INFINITY INFINITY INTUITIONS, PARADOXES, AND COGNITIVE LEAPS by by Ami M. Mamolo Mamolo M.Sc., McMaster University, 2005 2005 B.Sc., McMaster University, 2003 2003 THESIS SUBMITTED IN PARTIAL FULFILLMENT FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF OF DOCTOR OF PHILOSOPHY PHILOSOPHY In the Faculty Faculty of of Education Education © Ami M. Mamolo Mamolo SIMON FRASER UNIVERSITY UNIVERSITY Spring 2009 2009 All rights reserved. This work may not be be reproduced in whole or in part, by photocopy photocopy or other means without pennission ofthe author. author. APPROVAL Name: Ami Maria Mamolo Degree: Doctor of Philosophy Title of Thesis: Glimpses of Infinity: Intuitiol':ts, Paradoxes and Cognitive Leaps Examining Committee: Chair: Dr. Cecile Sabatier, Assistant Professor Dr. Rina Zazkis, Professor Senior Supervisor Dr. Peter Liljedahl, Assistant ProfessorProfessor Committee MemberMember Jason Bell, Assistant Professor, Department ofof MathematicsMathematics Committee MemberMember Dr. Nathalie Sinclair, Assistant Professor, Faculty ofof Education, Simon Fraser UniversityUniversity Internal/External ExaminerExaminer Dr. Peter Taylor, Professor, Department ofof Mathematics and Statistics, Queen's UniversityUniversity External ExaminerExaminer Date Defended/Approved: II SIMON FRASER UNIVERSITY LIBRARY Declaration of Partial Copyright Licence The dmhor, whose copynght is declared on the ntle pdge of rhis work, Ius gr,lllted w Sunon Fraser UniversIty the light to lend this thesis, project or extended essay to users of the Simon Fraser University Llbr;lry, dlld to m;lke pam,ll or single copiescaples only for such users or in respome to ,1 request froJll rhe library ofdny orher ulliver,lry, or other educational insritution, 011 its own behalf or for one of its users, The Hlthor has further granted pemussion ro SlI1lOn Fraser Un.iversiry ro keep or make ,1 digiral copy for use in its circulating collecrioncoll('crion (currently available to rhe public ar the "Insritutional Reposirory" link of rhe SFU Libl~lry website <V',;w\v.lib.sfu.ca> at: <lmp://ir.lib.sfu.ca/handJe/1892/112» ,Illd, wuhollt ch'lllging the coment. to tram],lte rhe thesls/pr~jeCl or exrended essays, if teChruClUy pos'lble,posslble, to ,In)' medium or [omlat for the purpose of preselv,1tion of rhe digit,ll \\lork. 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Simon Fraser Universlry Library Burnaby, BC, CanMb Revised Spring 2007 (SfU Library) SIMON !;RASER UNIVERSITY THINKING OF THE WORLO STATEMENT OF ETHICS APPROVAL The author, whose name appears on the title page of this work, has obtained. for the research described in this work, either: (a) Human research ethics approval from the Simon Fraser University Office of Research Ethics, or (b)(b) Advance approval of the animal care protocol from the University Animal Care Committee of Simon Fraser University; or has conducted the research (c)(c) as a co-investigator, in a research project approved in advance, or (dl as a member of a course approved in advance for minimal risk human research, by the Office of Research Ethics. A copy of the approval letter has been filed at the Theses Office of the University Library at the time of submission of this thesis or project. The originai application for approval and letter of approval are filed with the relevant offices. Inquiries may be directed to those authorities. Bennett Library Simon Fraser University Burnaby, BC, Canada laSl reVISion" Sumnler 2007 ABSTRACT This dissertation examines undergraduate and graduate university students' emergent conceptions of mathematical infinity. In particular, my research focuses on identifying the cognitive leaps required to overcome epistemological obstacles related to the idea of actual infinity. Extending on prior research regarding intuitive approaches to set comparison tasks, my research offers a refined analysis of the tacit conceptions and philosophies which influence learners' emergent understanding of mathematical infinity, as manifested through their engagement with geometric tasks and two well known paradoxes - Hilbert's Grand Hotel paradox and the Ping-Pong Ball Conundrum. In addition, my research sheds new light on specific features involved in accommodating the idea of actual infinity. The results of my research indicate that accommodating the idea of actual infinity requires a leap of imagination away from 'realistic' considerations and philosophical beliefs towards the 'realm of mathematics'. The abilities to clarify a separation between an intuitive and a fonnal understanding of infinity, and to conceive of 'infinite' as an answer to the question 'how many?' are also recognised as fundamental features III developing a nonnative understanding of actual infinity. Further, in order to accommodate the idea of actual infinity it is necessary to understand specific properties of transfinite arithmetic, in particular the indeterminacy of transfinite subtraction. III One afternoon I sat re{ax.jng on tfie grass) avoiding my responsi6irities.responsi6ifities. '}(ear6y, a group of cfiirdrencfiifdren prayed under tne stern eye of a cfiaperone. )Is tfie cfiifdrencfiirdren ran c{oser and c{oser to wfiere I sat) seeking ways to avoid tfieir cfiaperone) I 6ecame curious and ask,ed one cfiife!:cfiircf: "If you cou{d do anytfiing you wanted today, wfiat wou{d it 6e?)} erfie cfiifdcfiird rep{ied witfi a question of fiis own: "tyou mean 6eyond wfiat I cou{d actua{{y acfiieve? JJ *** erfiis dissertation is dedicated to tfie peop{e in my rife wfio fie{pedfiefped me acfiieve 6eyond wfiat I ever imagined, and wfiom I rove deady: ero my fami(y IV ACKNOWLEDGMENTS My sincerest thanks extend to Dr. Rina Zazkis. Her guidance, encouragement, and invaluable insight throughout the completion of my dissertation were a tremendous help and inspiration. 1 would like to extend my gratitude to Dr. Peter Liljedahl, whose valuable feedback and suggestions helped push my research and thinking into new directions, and to Dr. Jason Bell for his support, acumen, and humour. I would also like to express my appreciation to the participants of my studies, and to the Department of Mathematics at Simon Fraser University for their support and collegiality. v Table ofContents Approval Page ii ii Abstract iii iii Dedication iv iv Acknowledgements v v Table of Contents vi vi List of Figures ix ix Chapter I: Introduction 1 1 1. 1 History, Rationale, and Research Questions 1 1 1.2 Thesis Organisation 6 6 Chapter 22:: Infinity in Matbematics 8 8 2.1 Cantorian Set Theory 9 9 2.1.1 Transfinite Cardinals 10 10 2.1.2 Transfinite Ordinals 16 16 2.1.3 Cantor's Theorem and the Continuum Hypothesis l7 2.2 Nonstandard Analysis, Infinitesimals, and Calculus 19 19 2.2.1 Nonstandard Numbers 20 20 2.2.2 Infinitesimals and Limits 24 24 2.2.3 Series and Sequences 26 26 Chapter 33:: Paradoxes of the Infinite 28 28 3.1 Infinite Series, Finite Sums 30 30 3.1.1 The Dichotomy Paradox 30 30 3.1.2 Achilles and the Tortoise 31 31 3.1.3 Common Themes 32 32 3.2 Cardinality and Infinite Sets 33 33 3.2.1 Galileo's Paradox 34 34 3.2.2 Hilbert's Paradox: The Grand HotelHotel.... 36 36 3.2.3 Common Themes 36 36 3.3 Transfinite Subtraction 37 37 3.3.1 The Ping-Pong Ball Conundrum 38 38 3.3.2 The Ping-Pong Ball Variation 39 39 3.3.3 Common Themes 40 40 VI Chapter 44:: Infinity in Mathematics Education Research Research 42 42 4.1 Intuitions ofa Single, 'Endless' Infinite Infinite 43 43 4.2 Na"ive Resolutions to Set Comparison Tasks Tasks 45 45 4.3 Instruction and Mathematical Sophistication Sophistication 49 49 4.4 Pedagogical Strategies Strategies 51 51 Chapter 55:: Theoretical Perspectives Perspectives 55 55 5.1 Reducing Abstraction and Measuring Infinity Infinity 58 58 5.1.1 Measuring Infinity Infinity 58 58 5.1.2 Coping with the Unfamiliar Unfamiliar 59 59 5.2 APOS Theory Theory 61 61 5.2.1 Infinity as Process and ObjectObject...... 62 62 Chapter 66:: Intuitions of 'Infinite Numbers': Infinite Magnitude vs. Infinite Infinite Representation 65 65 6.1 Setting and Methodology Methodology 66 66 6.2 Results and Analysis Analysis 68 68 6.2.1 Bound Infinity: Numbers and Points Points 69 69 6.2.2 Subtracting Infinity: Intuition of 'Measuring Infinity' Infinity' 74 74 6.2.3 'Infinite Numbers': On-going Decimals Decimals 78 78 6.2.4 After Instruction: Robbie and Grace
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