DOCSLIB.ORG

Foundations of Geometry/Gerard A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Acquisitions Editor: Marnie Greenhut Assistant Editor: Elle Driska Senior Managing Editor: Karen Wernholm Production Project Manager: Beth Houston Marketing Manager: Roxanne McCarley Marketing Assistant: Caitlin Crane Senior Author Support/Technology Specialist: Joe Vetere Senior Manufacturing Buyer: Carol Melville Design Manager: Andrea Nix Senior Design Specialist: Heather Scott Cover Image Credit: Science and Society/Superstock Other photos courtesy of Gerard A. Venema Copyright © 2012, 2006 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm. Library of Congress Cataloging-in-Publication Data Venema, Gerard. The foundations of geometry/Gerard A. Venema. -- 2nd ed. p. cm. Includes bibliographical references index. ISBN 0-13-602058-5 1. Geometry—Foundations--Textbooks. I. Title. QA445.V455 2012 2012 514--dc22 2011002700 CIP 10 9 8 7 6 5 4 3 2 1 www.pearsonhighered.com ISBN-10: 0-13-602058-5 ISBN-13: 978-0-13-602058-5 Second Edition Foundations of Geometry Gerard A. Venema Department of Mathematics and Statistics Calvin College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo This page intentionally left blank To my family: Patricia, Sara, Emily, Daniel, David, Christian, and Ian. This page intentionally left blank Contents Preface ix 1 Prologue: Euclid’s Elements 1 1.1 Geometry before Euclid ............................. 1 1.2 The logical structure of Euclid’s Elements ................... 2 1.3 The historical significance of Euclid’s Elements ................ 3 1.4 A look at Book I of the Elements ........................ 4 1.5 A critique of Euclid’s Elements ......................... 7 1.6 Final observations about the Elements ..................... 9 2 Axiomatic Systems and Incidence Geometry 14 2.1 The structure of an axiomatic system ...................... 14 2.2 An example: Incidence geometry ........................ 16 2.3 The parallel postulates in incidence geometry ................. 20 2.4 Axiomatic systems and the real world ...................... 22 2.5 Theorems, proofs, and logic ........................... 24 2.6 Some theorems from incidence geometry .................... 32 3 Axioms for Plane Geometry 35 3.1 The undefined terms and two fundamental axioms .............. 36 3.2 Distance and the Ruler Postulate ........................ 37 3.3 Plane separation .................................. 46 3.4 Angle measure and the Protractor Postulate .................. 51 3.5 The Crossbar Theorem and the Linear Pair Theorem ............. 55 3.6 The Side-Angle-Side Postulate .......................... 62 3.7 The parallel postulates and models ....................... 66 4 Neutral Geometry 69 4.1 The Exterior Angle Theorem and existence of perpendiculars . ...... 70 4.2 Triangle congruence conditions ......................... 74 4.3 Three inequalities for triangles .......................... 77 4.4 The Alternate Interior Angles Theorem .................... 82 4.5 The Saccheri-Legendre Theorem ........................ 84 4.6 Quadrilaterals ................................... 88 4.7 Statements equivalent to the Euclidean Parallel Postulate .......... 91 4.8 Rectangles and defect ............................... 97 4.9 The Universal Hyperbolic Theorem .......................104 5 Euclidean Geometry 106 5.1 Basic theorems of Euclidean geometry .....................107 5.2 The Parallel Projection Theorem ........................110 5.3 Similar triangles ..................................112 5.4 The Pythagorean Theorem ............................114 5.5 Trigonometry ...................................116 5.6 Exploring the Euclidean geometry of the triangle ...............118 v vi Contents 6 Hyperbolic Geometry 131 6.1 Basic theorems of hyperbolic geometry .....................133 6.2 Common perpendiculars .............................138 6.3 The angle of parallelism .............................140 6.4 Limiting parallel rays ...............................144 6.5 Asymptotic triangles ...............................149 6.6 The classification of parallels ...........................152 6.7 Properties of the critical function ........................155 6.8 The defect of a triangle ..............................160 6.9 Is the real world hyperbolic? ...........................164 7 Area 167 7.1 The Neutral Area Postulate ...........................167 7.2 Area in Euclidean geometry ...........................173 7.3 Dissection theory .................................180 7.4 Proof of the dissection theorem in Euclidean geometry ............181 7.5 The associated Saccheri quadrilateral ......................185 7.6 Area and defect in hyperbolic geometry ....................190 8 Circles 195 8.1 Circles and lines in neutral geometry ......................195 8.2 Circles and triangles in neutral geometry ....................200 8.3 Circles in Euclidean geometry ..........................206 8.4 Circular continuity .................................212 8.5 Circumference and area of Euclidean circles ..................215 8.6 Exploring Euclidean circles ............................222 9 Constructions 228 9.1 Compass and straightedge constructions ....................228 9.2 Neutral constructions ...............................231 9.3 Euclidean constructions ..............................234 9.4 Construction of regular polygons ........................235 9.5 Area constructions ................................239 9.6 Three impossible constructions .........................242 10 Transformations 245 10.1 Properties of isometries ..............................246 10.2 Rotations, translations, and glide reflections ..................252 10.3 Classification of Euclidean motions .......................260 10.4 Classification of hyperbolic motions .......................263 10.5 A transformational approach to the foundations ................265 10.6 Similarity transformations in Euclidean geometry ...............270 10.7 Euclidean inversions in circles ..........................275 11 Models 287 11.1 The Cartesian model for Euclidean geometry .................289 11.2 The Poincare´ disk model for hyperbolic geometry ...............290 11.3 Other models for hyperbolic geometry .....................296 11.4 Models for elliptic geometry ...........................301 12 Polygonal Models and the Geometry of Space 303 12.1 Curved surfaces ..................................304 Contents vii 12.2 Approximate models for the hyperbolic plane .................315 12.3 Geometric surfaces ................................321 12.4 The geometry of the universe ..........................326 12.5 Conclusion .....................................331 12.6 Further study ....................................331 12.7 Templates .....................................336 APPENDICES A Euclid’s Book I 344 A.1 Definitions .....................................344 A.2 Postulates ......................................346 A.3 Common Notions .................................346 A.4 Propositions ....................................346 B Systems of Axioms for Geometry 350 B.1 Hilbert’s axioms ..................................351 B.2 Birkhoff’s axioms .................................353 B.3 MacLane’s axioms .................................354 B.4 SMSG axioms ...................................355 B.5 UCSMP axioms ..................................357 C The Postulates Used in this Book 360 C.1 Criteria used in selecting the postulates .....................360 C.2 Statements of the postulates ...........................362 C.3 Logical relationships ...............................363 D The van Hiele Model 365 E Set Notation and the Real Numbers 366 E.1 Some elementary set theory ...........................366 E.2 Axioms for the real numbers ...........................367 E.3 Properties of the real numbers ..........................368 E.4 One-to-one and onto functions ..........................370 E.5 Continuous functions ...............................371 F Hints for Selected Exercises 372 Bibliography 379 Index 382 This page intentionally left blank Preface This is a textbook for an undergraduate course in axiomatic geometry. The text is targeted at mathematics students who have completed the calculus sequence and perhaps a first course in linear algebra, but who have not yet encountered such upper-level mathematics courses as real analysis and abstract algebra. A course based on this book will enrich the education of all mathematics majors and will ease their transition into more advanced mathematics courses. The book also includes emphases that make it especially appropriate as the textbook for a geometry course taken by future high school mathematics teachers. WHAT’S NEW IN THE SECOND EDITION? For the benefit of those who have used the first edition of this book, here is a quick summary of what has changed in the second edition. • The first part
Recommended publications
  • Theorems and Postulates

    Theorems and Postulates

    Theorems and Postulates JJJG Given AB and a number r between 0 and 180, there Postulate 1-A is exactly one JrayJJG with endpoint A , extending on Protractor Postulate either side of AB , such that the measure of the angle formed is r . ∠A is a right angle if mA∠ is 90. Definition of Right, Acute ∠A is an acute angle if mA∠ is less than 90. and Obtuse Angles ∠A is an obtuse angle if mA∠ is greater than 90 and less than 180. Postulate 1-B If R is in the interior of ∠PQS , then mPQRmRQSmPQS∠ +∠ =∠ . Angle Addition If mP∠+∠=∠QRmRQSmPQS, then R is in the interior of ∠PQS. Vertical angles are congruent. The sum of the measures of the angles in a linear pair is 180˚. The sum of the measures of complementary angles is 90˚. Two points on a line can be paired with real numbers so that, given Postulate 2-A any two points R and S on the line, R corresponds to zero, and S Ruler corresponds to a positive number. Point R could be paired with 0, and S could be paired with 10. R S -2 -1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 Postulate 2-B If N is between M and P, then MN + NP = MP. Segment Addition Conversely, if MN + NP = MP, then N is between M and P. Theorem 2-A In a right triangle, the sum of the squares of the measures Pythagorean of the legs equals the square of the measure of the Theorem hypotenuse.
  • David Hilbert's Contributions to Logical Theory

    David Hilbert's Contributions to Logical Theory

    David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques.
  • Read Book Advanced Euclidean Geometry Ebook

    Read Book Advanced Euclidean Geometry Ebook

    ADVANCED EUCLIDEAN GEOMETRY PDF, EPUB, EBOOK Roger A. Johnson | 336 pages | 30 Nov 2007 | Dover Publications Inc. | 9780486462370 | English | New York, United States Advanced Euclidean Geometry PDF Book As P approaches nearer to A , r passes through all values from one to zero; as P passes through A , and moves toward B, r becomes zero and then passes through all negative values, becoming —1 at the mid-point of AB. Uh-oh, it looks like your Internet Explorer is out of date. In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Calculus Real analysis Complex analysis Differential equations Functional analysis Harmonic analysis. This article needs attention from an expert in mathematics. Facebook Twitter. On any line there is one and only one point at infinity. This may be formulated and proved algebraically:. When we have occasion to deal with a geometric quantity that may be regarded as measurable in either of two directions, it is often convenient to regard measurements in one of these directions as positive, the other as negative. Logical questions thus become completely independent of empirical or psychological questions For example, proposition I. This volume serves as an extension of high school-level studies of geometry and algebra, and He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.
  • The Stoics and the Practical: a Roman Reply to Aristotle

    The Stoics and the Practical: a Roman Reply to Aristotle

    DePaul University Via Sapientiae College of Liberal Arts & Social Sciences Theses and Dissertations College of Liberal Arts and Social Sciences 8-2013 The Stoics and the practical: a Roman reply to Aristotle Robin Weiss DePaul University, [email protected] Follow this and additional works at: https://via.library.depaul.edu/etd Recommended Citation Weiss, Robin, "The Stoics and the practical: a Roman reply to Aristotle" (2013). College of Liberal Arts & Social Sciences Theses and Dissertations. 143. https://via.library.depaul.edu/etd/143 This Thesis is brought to you for free and open access by the College of Liberal Arts and Social Sciences at Via Sapientiae. It has been accepted for inclusion in College of Liberal Arts & Social Sciences Theses and Dissertations by an authorized administrator of Via Sapientiae. For more information, please contact [email protected]. THE STOICS AND THE PRACTICAL: A ROMAN REPLY TO ARISTOTLE A Thesis Presented in Partial Fulfillment of the Degree of Doctor of Philosophy August, 2013 BY Robin Weiss Department of Philosophy College of Liberal Arts and Social Sciences DePaul University Chicago, IL - TABLE OF CONTENTS - Introduction……………………..............................................................................................................p.i Chapter One: Practical Knowledge and its Others Technê and Natural Philosophy…………………………….....……..……………………………….....p. 1 Virtue and technical expertise conflated – subsequently distinguished in Plato – ethical knowledge contrasted with that of nature in
  • Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg

    Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg

    Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By “elementary” plane geometry I mean the geometry of lines and circles—straight- edge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incom- pleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry. 1. AXIOMATIC DEVELOPMENT 1.0. Viewpoint. Euclid’s Elements was the first axiomatic presentation of mathemat- ics, based on his five postulates plus his “common notions.” It wasn’t until the end of the nineteenth century that rigorous revisions of Euclid’s axiomatics were presented, filling in the many gaps in his definitions and proofs. The revision with the great- est influence was that by David Hilbert starting in 1899, which will be discussed below. Hilbert not only made Euclid’s geometry rigorous, he investigated the min- imal assumptions needed to prove Euclid’s results, he showed the independence of some of his own axioms from the others, he presented unusual models to show certain statements unprovable from others, and in subsequent editions he explored in his ap- pendices many other interesting topics, including his foundation for plane hyperbolic geometry without bringing in real numbers.
  • Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used

    Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used

    Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Definitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent.
  • Foundations of Geometry

    Foundations of Geometry

    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.
  • Mechanism, Mentalism, and Metamathematics Synthese Library

    Mechanism, Mentalism, and Metamathematics Synthese Library

    MECHANISM, MENTALISM, AND METAMATHEMATICS SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Florida State University Editors: ROBER T S. COHEN, Boston University DONALD DAVIDSON, University o/Chicago GABRIEL NUCHELMANS, University 0/ Leyden WESLEY C. SALMON, University 0/ Arizona VOLUME 137 JUDSON CHAMBERS WEBB Boston University. Dept. 0/ Philosophy. Boston. Mass .• U.S.A. MECHANISM, MENT ALISM, AND MET AMA THEMA TICS An Essay on Finitism i Springer-Science+Business Media, B.V. Library of Congress Cataloging in Publication Data Webb, Judson Chambers, 1936- CII:J Mechanism, mentalism, and metamathematics. (Synthese library; v. 137) Bibliography: p. Includes indexes. 1. Metamathematics. I. Title. QA9.8.w4 510: 1 79-27819 ISBN 978-90-481-8357-9 ISBN 978-94-015-7653-6 (eBook) DOl 10.1007/978-94-015-7653-6 All Rights Reserved Copyright © 1980 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1980. Softcover reprint of the hardcover 1st edition 1980 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS PREFACE vii INTRODUCTION ix CHAPTER I / MECHANISM: SOME HISTORICAL NOTES I. Machines and Demons 2. Machines and Men 17 3. Machines, Arithmetic, and Logic 22 CHAPTER II / MIND, NUMBER, AND THE INFINITE 33 I. The Obligations of Infinity 33 2. Mind and Philosophy of Number 40 3. Dedekind's Theory of Arithmetic 46 4.
  • Definitions and Nondefinability in Geometry 475 2

    Definitions and Nondefinability in Geometry 475 2

    Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal.
  • The Foundations of Geometry

    The Foundations of Geometry

    The Foundations of Geometry Gerard A. Venema Department of Mathematics and Statistics Calvin College SUB Gottingen 7 219 059 926 2006 A 7409 PEARSON Prentice Hall Upper Saddle River, New Jersey 07458 Contents Preface ix Euclid's Elements 1 1.1 Geometry before Euclid 1 1.2 The logical structure of Euclid's Elements 2 1.3 The historical importance of Euclid's Elements 3 1.4 A look at Book I of the Elements 5 1.5 A critique of Euclid's Elements 8 1.6 Final observations about the Elements 11 Axiomatic Systems and Incidence Geometry 17 2.1 Undefined and defined terms 17 2.2 Axioms 18° 2.3 Theorems 18 2.4 Models 19 2.5 An example of an axiomatic system 19 2.6 The parallel postulates 25 2.7 Axiomatic systems and the real world 27 Theorems, Proofs, and Logic 31 3.1 The place of proof in mathematics 31 3.2 Mathematical language 32 3.3 Stating theorems 34 3.4 Writing proofs 37 3.5 Indirect proof \. 39 3.6 The theorems of incidence geometry ';, . 40 Set Notation and the Real Numbers 43 4.1 Some elementary set theory 43 4.2 Properties of the real numbers 45 4.3 Functions ,48 4.4 The foundations of mathematics 49 The Axioms of Plane Geometry 52 5.1 Systems of axioms for geometry 53 5.2 The undefined terms 56 5.3 Existence and incidence 56 5.4 Distance 57 5.5 Plane separation 63 5.6 Angle measure 67 vi Contents 5.7 Betweenness and the Crossbar Theorem 70 5.8 Side-Angle-Side 84 5.9 The parallel postulates 89 5.10 Models 90 6 Neutral Geometry 94 6.1 Geometry without the parallel postulate 94 6.2 Angle-Side-Angle and its consequences 95 6.3 The Exterior
  • Foundations of Mathematics

    Foundations of Mathematics

    Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions .
  • Chapter 1 the Foundation of Euclidean Geometry

    Chapter 1 the Foundation of Euclidean Geometry

    Chapter 1 The Foundation of Euclidean Geometry \This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state." | Clifford 1. Introduction. Geometry, that branch of mathematics in which are treated the properties of figures in space, is of ancient origin. Much of its development has been the result of efforts made throughout many centuries to construct a body of logical doctrine for correlating the geometrical data obtained from observation and measurement. By the time of Euclid (about 300 B.C.) the science of geometry had reached a well-advanced stage. From the accumulated material Euclid compiled his Elements, the most remarkable textbook ever written, one which, despite a number of grave imperfections, has served as a model for scientific treatises for over two thousand years. Euclid and his predecessors recognized what every student of philosophy knows: that not everything can be proved. In building a logical structure, one or more of the propositions must be assumed, the others following by logical deduction. Any attempt to prove all of the propositions must lead inevitably to the completion of a vicious circle. In geometry these assumptions originally took the form of postulates suggested by experience and intuition. At best these were statements of what seemed from observation to be true or approximately true. A geometry carefully built upon such a foundation may be expected to correlate the data of observation very well, perhaps, but certainly not exactly. Indeed, it should be clear that the mere change of some more-or-less doubtful postulate of one geometry may lead to another geometry which, although radically different from the first, relates the same data quite as well.