DOCSLIB.ORG

Revision V2.0, Chapter I Foundations of Geometry in the Plane

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Revision V2.0, Chapter I Foundations of Geometry in the Plane Revision v2.0 , Chapter I Foundations of Geometry in the Plane The primary goal of this book is to display the richness, unity, and insight that the concept of symmetry brings to the study of geometry. To do so, we must start from a solid grounding in elementary geometry. This chapter provides that necessary background. A rigorous treatment of foundational geometry is hardly elementary: it re- quires a careful development of appropriate concepts, axioms, and theorems. We will not give all the details for such a full axiomatic treatment. We will, however, structure the discussion in this chapter to outline such a treatment. The logical clarity of this approach helps in understanding and remembering the material.1 There are various ways to build a foundation for geometry. Our approach is that of metric geometry, which means that distance between points and angular measure are central organizing concepts. These concepts, in turn, are dependent on the basic properties of R, the real number system. We thus begin our discussion with some comments about the real numbers. I.1 The Real Numbers § We denote the set of real numbers with the symbol R. Other important number systems are Z, the set of integers; N, the set of natural numbers (non-negative integers); and Q, the set of rational numbers (quotients of integers). Later we will also discuss C, the set of complex numbers. Intuitively the real numbers correspond to the points on a straight line. Given a line ℓ, fix a unit of length (e.g., an inch), designate a point on O ℓ as the origin, and specify the “positive” and “negative” sides of ℓ. Then every point p of ℓ respresents one unique real number xp, where xp is the distance of p to (positive or negative depending on the side of ) — this is O O shown in Figure 1.1. The algebraic operations of addition and multiplication of real numbers then correspond to geometric movements on the line. 1We closely follow the development of geometry found in Edwin Moise, Elementary Geometry from an Advanced Standpoint, Third Edition, Addison Wesley Publishers. This book is the primary reference for the basic geometry to be used in Continuous Symmetry. 1 2 Revision v2.0 , Chapter I, Foundations of Geometry in the Plane distance x p p O ℓ -2 -1 0 1xp 2 Figure 1.1. The real number line. Using the number line, the order operation on real numbers, x<y, read “x is less than y,” means that the point corresponding to x occurs “to the left” of the point corresponding to y. The inequality x y, read “x is less than ≤ or equal to y,” means either x<y or x = y. We will assume all the standard algebraic properties of the real numbers, i.e., all the standard properties of addition, multiplication, subtraction, division, and order. For example, if x + r = y + r, then x = y, or if x<y and r > 0, then rx < ry. Do not underestimate the importance or the depth of this assumption! We base our development of geometry on the real number system, a system whose existence is non-trivial to establish and whose prop- erties are sophisticated. A proper study of the real number system belongs to the mathematical subject known as analysis. Using the order relations, we define various types of intervals for real num- bers a < b: bounded closed interval: [a, b]= all x such that a x b , { ≤ ≤ } bounded open interval: (a, b)= all x such that a<x<b . { } We allow open endpoints with a = or b = . However, in those cases −∞ ∞ the intervals are unbounded. The real numbers include all the natural numbers 1, 2, 3, . An important property of this inclusion is called the Archimedean Ordering Principle: For any real number x there exists a natural number n greater than x. One simple but highly important consequence of this property is that for any positive real number ǫ> 0, no matter how small, there exists a natural number n such that 1/n < ǫ (Exercise 1.1). Thus the fraction 1/n can be made “arbitrarily small” by choosing n sufficiently large. Completeness of the Real Number System. An intuitive understand- ing of real numbers as developed in, say, advanced secondary school algebra, will suffice for most of our work. However, there is one far deeper fact that we will need at certain times: the real number line “has no holes.” When rig- orously formulated, this property is known as the completeness of the real number system. Some readers may wish to defer this sophisticated concept until needed later in the text. Such readers should now skip to I.2. § §I.1. The Real Numbers 3 There are various equivalent ways to describe completeness — we will give a description that is easy to picture using closed intervals. A sequence of bounded, closed intervals [a1, b1], [a2, b2], . is said to be nested if each interval contains the next one as a subset. Written in set notation, this means [a , b ] [a , b ] [an, bn] . 1 1 ⊇ 2 2 ⊇···⊇ ⊇··· The real number system R is said to be complete because every such nested sequence of bounded closed intervals has at least one real number x that belongs to all the intervals: Theorem 1.2. The Completeness of R. For any nested sequence of bounded closed intervals in R there will al- ways be at least one real number x that belongs to all the intervals (i.e., x is in the intersection of all the intervals). This theorem is illustrated in Figure 1.3. [a1, b1] [a2, b2] ··· [an , bn ] ··· ··· ··· ℓ a1 a2 an x b b2 b1 ············n Figure 1.3. All the nested intervals [an, bn], n = 1, 2, 3,... , contain x. If the nested intervals are not closed, then there might or might not be a point common to all the intervals. This is examined in Exercise 1.3. The intuitive meaning of completeness is that the real number line “has no holes.” If you have not studied a rigorous formulation of the real number system (such as in an introductory analysis course), it’s probably hard to appreciate the importance of completeness. In fact, it is of critical impor- tance in much of mathematics — it guarantees that real numbers exist when and where we need them. An example of a number system which is not com- plete is Q, the set of rational numbers (fractions). It would, for example, be very difficult to develop calculus with just rational numbers — there are too many “holes” because the rationals are not complete. These ideas are more fully explored in the exercises. More on Completeness: Bolzano’s Theorem. There are several prop- erties of the real number system that are implied by completeness. One such property we will use later in the book is every bounded sequence in R has a convergent subsequence. This is known as Bolzano’s Theorem. We now explain this result. 4 Revision v2.0 , Chapter I, Foundations of Geometry in the Plane ∞ A sequence xn of real numbers is termed bounded if all the elements { }n=1 of the sequence are contained in a bounded interval, i.e., there exists a ∞ bounded interval [a, b] such that a xn b for all n. A sequence xn ≤ ≤ { }n=1 is said to converge to x if the terms in the sequence become arbitrarily close to x as the index n becomes large. A sequence can be bounded but not converge: a simple example is the sequence 1, 1, 1, 1, 1, 1,... { − − − } However, this sequence does have convergent subsequences; one example is 1, 1, 1,... In fact, the completeness of the real number system im- { } plies that every bounded sequence has a convergent subsequence. This is Bolzano’s Theorem. Theorem 1.4. Bolzano’s Theorem. Every bounded sequence in R has a convergent subsequence. The proof of Bolzano’s Theorem, showing it to be a consequence of the completeness of the real number system as formulated in Theorem 1.2, is given in 14. § Exercises I.1 “Why,” said the Dodo, “the best way to explain it is to do it.” Lewis Carroll Exercise 1.1. (a) Show that for any positive real number ǫ> 0, no matter how small, there exists a natural number n such that 1/n<ǫ. Hint: Use the Archimedean Ordering Principle with x = 1/ǫ. (b) For any real number x′ show there exists an integer n′ less than x′. Hint: Use the Archimedean Ordering Principle with x = x′. − Exercise 1.2. (a) Given any real number x, show there exists a smallest integer n0 such that x<n0. Hint: Use Archimedean order to show there exists an integer N such that x < N. Then use Exercise 1.1b to show that the set of integers k x < k N is non-empty and { | ≤ } finite. Any finite set of real numbers must have a smallest member. (b) Suppose x1 and x2 are two real numbers such that x1 <x2. Prove there exists a rational number r = n/m such that x1 <r<x2. Outline: First show there exists a positive integer m large enough so that 1/m < x x . Then show there exists a smallest integer 2 − 1 n such that x1 < n/m. To verify that r = n/m is what you need, you must only verify r<x2. (c) Suppose x1 and x2 are two real numbers such that whenever r1 and r2 are rational numbers satisfying r1 < x1 < r2, then r1 < x2 < r2 §I.1.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]