Cantorian Abstraction: a Reconstruction and Defence
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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
THE THEORY of MATHEMATICAL SUBTRACTION in ARISTOTLE By
THE THEORY OF MATHEMATICAL SUBTRACTION IN ARISTOTLE by Ksenia Romashova Submitted in partial fulfilment of the requirements for the degree of Master of Arts at Dalhousie University Halifax, Nova Scotia August 2019 © Copyright by Ksenia Romashova, 2019 DEDICATION PAGE To my Mother, Vera Romashova Мама, Спасибо тебе за твою бесконечную поддержку, дорогая! ii TABLE OF CONTENTS LIST OF FIGURES .................................................................................................................. iv ABSTRACT ............................................................................................................................... v LIST OF ABBREVIATIONS USED ....................................................................................... vi ACKNOWLEDGEMENTS ..................................................................................................... vii CHAPTER 1 INTRODUCTION ............................................................................................... 1 CHAPTER 2 THE MEANING OF ABSTRACTION .............................................................. 6 2.1 Etymology and Evolution of the Term ................................................................... 7 2.2 The Standard Phrase τὰ ἐξ ἀφαιρέσεως or ‘Abstract Objects’ ............................ 19 CHAPTER 3 GENERAL APPLICATION OF ABSTRACTION .......................................... 24 3.1 The Instances of Aphairein in Plato’s Dialogues ................................................. 24 3.2 The Use of Aphairein in Aristotle’s Topics -
Continuity and Mathematical Ontology in Aristotle
Journal of Ancient Philosophy, vol. 14 issue 1, 2020. DOI: http://dx.doi.org/10.11606/issn.1981-9471.v14i1p30-61 Continuity and Mathematical Ontology in Aristotle Keren Wilson Shatalov In this paper I argue that Aristotle’s understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle’s notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle’s understanding of continuity can only be saved if he holds a certain kind of mathematical ontology. My topic in this paper is Aristotle’s understanding of mathematical continuity. While the idea that continua are composed of infinitely many points is the present day orthodoxy, the Aristotelian understanding of continua as non-punctiform and infinitely divisible was the reigning theory for much of the history of western mathematics, and there is renewed interest in it from current mathematicians and philosophers of mathematics. -
Formal Construction of a Set Theory in Coq
Saarland University Faculty of Natural Sciences and Technology I Department of Computer Science Masters Thesis Formal Construction of a Set Theory in Coq submitted by Jonas Kaiser on November 23, 2012 Supervisor Prof. Dr. Gert Smolka Advisor Dr. Chad E. Brown Reviewers Prof. Dr. Gert Smolka Dr. Chad E. Brown Eidesstattliche Erklarung¨ Ich erklare¨ hiermit an Eides Statt, dass ich die vorliegende Arbeit selbststandig¨ verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverstandniserkl¨ arung¨ Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit vero¨ffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrucken,¨ (Datum/Date) (Unterschrift/Signature) iii Acknowledgements First of all I would like to express my sincerest gratitude towards my advisor, Chad Brown, who supported me throughout this work. His extensive knowledge and insights opened my eyes to the beauty of axiomatic set theory and foundational mathematics. We spent many hours discussing the minute details of the various constructions and he taught me the importance of mathematical rigour. Equally important was the support of my supervisor, Prof. Smolka, who first introduced me to the topic and was there whenever a question arose. -
Berkeley's Case Against Realism About Dynamics
Lisa Downing [Published in Berkeley’s Metaphysics, ed. Muehlmann, Penn State Press 1995, 197-214. Turbayne Essay Prize winner, 1992.] Berkeley's case against realism about dynamics While De Motu, Berkeley's treatise on the philosophical foundations of mechanics, has frequently been cited for the surprisingly modern ring of certain of its passages, it has not often been taken as seriously as Berkeley hoped it would be. Even A.A. Luce, in his editor's introduction to De Motu, describes it as a modest work, of limited scope. Luce writes: The De Motu is written in good, correct Latin, but in construction and balance the workmanship falls below Berkeley's usual standards. The title is ambitious for so brief a tract, and may lead the reader to expect a more sustained argument than he will find. A more modest title, say Motion without Matter, would fitly describe its scope and content. Regarded as a treatise on motion in general, it is a slight and disappointing work; but viewed from a narrower angle, it is of absorbing interest and high importance. It is the application of immaterialism to contemporary problems of motion, and should be read as such. ...apart from the Principles the De Motu would be nonsense.1 1The Works of George Berkeley, Bishop of Cloyne, ed. A.A. Luce and T.E. Jessop (London: Thomas Nelson and Sons, 1948-57), 4: 3-4. In this paper, all references to Berkeley are to the Luce-Jessop edition. Quotations from De Motu are taken from Luce's translation. I use the following abbreviations for Berkeley’s works: PC Philosophical Commentaries PHK-I Introduction to The Principles of Human Knowledge PHK The Principles of Human Knowledge DM De Motu A Alciphron TVV The Theory of Vision Vindicated and Explained S Siris 1 There are good general reasons to think, however, that Berkeley's aims in writing the book were as ambitious as the title he chose. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Frege and the Logic of Sense and Reference
FREGE AND THE LOGIC OF SENSE AND REFERENCE Kevin C. Klement Routledge New York & London Published in 2002 by Routledge 29 West 35th Street New York, NY 10001 Published in Great Britain by Routledge 11 New Fetter Lane London EC4P 4EE Routledge is an imprint of the Taylor & Francis Group Printed in the United States of America on acid-free paper. Copyright © 2002 by Kevin C. Klement All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any infomration storage or retrieval system, without permission in writing from the publisher. 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Klement, Kevin C., 1974– Frege and the logic of sense and reference / by Kevin Klement. p. cm — (Studies in philosophy) Includes bibliographical references and index ISBN 0-415-93790-6 1. Frege, Gottlob, 1848–1925. 2. Sense (Philosophy) 3. Reference (Philosophy) I. Title II. Studies in philosophy (New York, N. Y.) B3245.F24 K54 2001 12'.68'092—dc21 2001048169 Contents Page Preface ix Abbreviations xiii 1. The Need for a Logical Calculus for the Theory of Sinn and Bedeutung 3 Introduction 3 Frege’s Project: Logicism and the Notion of Begriffsschrift 4 The Theory of Sinn and Bedeutung 8 The Limitations of the Begriffsschrift 14 Filling the Gap 21 2. The Logic of the Grundgesetze 25 Logical Language and the Content of Logic 25 Functionality and Predication 28 Quantifiers and Gothic Letters 32 Roman Letters: An Alternative Notation for Generality 38 Value-Ranges and Extensions of Concepts 42 The Syntactic Rules of the Begriffsschrift 44 The Axiomatization of Frege’s System 49 Responses to the Paradox 56 v vi Contents 3. -
A Functional Hitchhiker's Guide to Hereditarily Finite Sets, Ackermann
A Functional Hitchhiker’s Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions – unpublished draft – Paul Tarau Department of Computer Science and Engineering University of North Texas [email protected] Abstract interest in various fields, from representing structured data The paper is organized as a self-contained literate Haskell in databases (Leontjev and Sazonov 2000) to reasoning with program that implements elements of an executable fi- sets and set constraints in a Logic Programming framework nite set theory with focus on combinatorial generation and (Dovier et al. 2000; Piazza and Policriti 2004; Dovier et al. arithmetic encodings. The code, tested under GHC 6.6.1, 2001). is available at http://logic.csci.unt.edu/tarau/ The Universe of Hereditarily Finite Sets is built from the research/2008/fSET.zip. empty set (or a set of Urelements) by successively applying We introduce ranking and unranking functions generaliz- powerset and set union operations. A surprising bijection, ing Ackermann’s encoding to the universe of Hereditarily Fi- discovered by Wilhelm Ackermann in 1937 (Ackermann nite Sets with Urelements. Then we build a lazy enumerator 1937; Abian and Lamacchia 1978; Kaye and Wong 2007) for Hereditarily Finite Sets with Urelements that matches the from Hereditarily Finite Sets to Natural Numbers, was the unranking function provided by the inverse of Ackermann’s original trigger for our work on building in a mathematically encoding and we describe functors between them resulting elegant programming language, a concise and executable in arithmetic encodings for powersets, hypergraphs, ordinals hereditarily finite set theory. The arbitrary size of the data and choice functions. -
Ordinal Numbers and the Well-Ordering Theorem Ken Brown, Cornell University, September 2013
Mathematics 4530 Ordinal numbers and the well-ordering theorem Ken Brown, Cornell University, September 2013 The ordinal numbers form an extension of the natural numbers. Here are the first few of them: 0; 1; 2; : : : ; !; ! + 1;! + 2;:::;! + ! =: !2;!2 + 1;:::;!3;:::;!2;:::: They go on forever. As soon as some initial segment of ordinals has been con- structed, a new one is adjoined that is bigger than all of them. The theory of ordinals is closely related to the theory of well-ordered sets (see Section 10 of Munkres). Recall that a simply ordered set X is said to be well- ordered if every nonempty subset Y has a smallest element, i.e., an element y0 such that y0 ≤ y for all y 2 Y . For example, the following sets are well-ordered: ;; f1g ; f1; 2g ; f1; 2; : : : ; ng ; f1; 2;::: g ; f1; 2;:::;!g ; f1; 2; : : : ; !; ! + 1g : On the other hand, Z is not well-ordered in its natural ordering. As the list of examples suggests, constructing arbitrarily large well-ordered sets is essentially the same as constructing the system of ordinal numbers. Rather than taking the time to develop the theory of ordinals, we will concentrate on well-ordered sets in what follows. Notation. In dealing with ordered sets in what follows we will often use notation such as X<x := fy 2 X j y < xg : A set of the form X<x is called a section of X. Well-orderings are useful because they allow proofs by induction: Induction principle. Let X be a well-ordered set and let Y be a subset. -
A Theory of Particular Sets, As It Uses Open Formulas Rather Than Sequents
A Theory of Particular Sets Paul Blain Levy, University of Birmingham June 14, 2019 Abstract ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sen- tences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets. 1 Introduction ZFC has long been established as a set-theoretic foundation of mathematics, but concerns about its meaningfulness have often been raised. Specifically, its use of unrestricted quantifiers seems to presuppose an absolute totality of sets. The purpose of this paper is to present a new set theory called TOPS that addresses this concern by speaking only about particular sets. Though weaker than ZFC, it is adequate for a large amount of mathematics. To explain the need for a new theory, we begin in Section 2 by surveying some basic mathematical beliefs and conceptions. Section 3 presents the language of TOPS, and Section 4 the theory itself, using Natural Deduction. Section 5 adapts the theory to allow theorems with free variables. Related work is discussed in Section 6. While TOPS is by no means the first set theory to use restricted quantifiers in order to avoid assuming a totality of sets, it is the first to do so using Natural Deduction, which turns out to be very helpful. This is apparent when TOPS is compared to a previously studied system [26] that proves essentially the same sentences but includes an inference rule that (we argue) cannot be considered truth-preserving. -
Sets As Graphs
Universita` degli Studi di Udine Dipartimento di Matematica e Informatica Dottorato di Ricerca in Informatica XXIV Ciclo - Anno Accademico 2011-2012 Ph.D. Thesis Sets as Graphs Candidate: Supervisors: Alexandru Ioan Tomescu Alberto Policriti Eugenio G. Omodeo December 2011 Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Author's e-mail: [email protected] Author's address: Dipartimento di Matematica e Informatica Universit`adegli Studi di Udine Via delle Scienze, 206 33100 Udine Italia Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine To my parents Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Abstract In set theory and formal logic, a set is generally an object containing nothing but other sets as elements. Not only sets enable uniformity in the formalization of the whole of mathematics, but their ease-of-use and conciseness are employed to represent information in some computer languages. Given the intrinsic nesting property of sets, it is natural to represent them as directed graphs: vertices will stand for sets, while the arc relation will mimic the membership relation. This switch of perspective is important: from a computational point of view, this led to many decidability results, while from a logical point of view, this allowed for natural extensions of the concept of set, such as that of hyperset. Interpreting a set as a directed graph gives rise to many combinatorial, structural and computational questions, having as unifying goal that of a transfer of results and techniques across the two areas. -
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University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/46969 This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. Supporting the Migration from Construal to Program: Rethinking Software Development by Nicolas William Pope Thesis Submitted to the University of Warwick for the degree of Doctor of Philosophy Department of Computer Science August 2011 Contents List of Tables vii List of Figures viii Acknowledgments xvi Declarations xvii Abstract xviii Abbreviations xix Chapter 1 Introduction 1 1.1 Plastic Applications . .1 1.2 Plastic Software Environments . .4 1.3 A Lack of Plasticity . .6 1.4 Thesis Aims . .9 1.5 Thesis Outline . 10 Chapter 2 Background 13 2.1 End-User Development . 13 2.1.1 Principles . 14 2.1.2 Common Approaches . 17 ii 2.1.3 Existing Environments . 17 2.1.4 Guidelines . 24 2.2 Empirical Modelling . 26 2.2.1 What is Empirical Modelling? . 26 2.2.2 The Principles . 32 2.2.3 Current Tools . 35 2.2.4 An Example Model . 38 2.2.5 EM and Software Development . 41 2.3 Miscellaneous Technologies . 43 Chapter 3 Enabling Plastic Applications 45 3.1 Empirical Modelling and Plastic Applications . 45 3.2 Dimensions of Refinement . 48 3.3 Limitations of EM Tools and Concepts .