Paradoxes in Logic and Set Theory
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Paradox Philosophy
Paradox Philosophy Course at Akita International University Philipp Blum, University of Lucerne, [email protected] https://philipp.philosophie.ch/teaching/paradoxes20.html Summary of the course, version of February 17, 2020 Contents 1 Administrative info 2 2 What is a paradox? 3 3 Mathematical Paradoxes 4 3.1 Things and their groupings .............................. 4 3.2 Cantor and modern set theory ............................ 4 3.3 Frege’s definition of number and Russell’s paradox ................. 6 4 Physical Paradoxes 7 4.1 Zeno’s paradoxes .................................... 7 4.2 Not easily solved by the modern calculus ....................... 9 4.3 Whitehead’s lesson: becoming is not continuous ................... 9 4.4 Russell’s lesson: the at-at theory of motion ..................... 10 4.5 Black’s lesson: the problem of hypertasks ...................... 11 4.6 Background 1: the line and the points ........................ 11 4.7 Background 2: the metaphysics of persistence and the problem of change . 12 5 Semantic Paradoxes 13 5.1 Truth and Falsity .................................... 13 5.2 The Liar ......................................... 14 5.3 The Strengthened Liar ................................. 15 5.4 A Spicy Curry ..................................... 15 5.5 Having fun with liars and curries ........................... 16 5.6 The Sorites paradox .................................. 17 6 Paradoxes of Rationality 18 6.1 Surprise Exam ..................................... 18 6.2 Prisoners’ Dilemma .................................. -
Barbers & Big Ideas: Paradox in Math and Poetry
Barbers & Big Ideas: Paradox in Math and Poetry by Alice Major Published in the Journal of Mathematics and the Arts Volume 8, Issue 1-2, 2014 Special Issue: Poetry and Mathematics The final version of this paper is Available at Taylor & Francis: http://dx.doi.org/10.1080/17513472.2014.943999 Abstract Paradox intrigues both mathematicians and artists of all kinds. Throughout the recorded history of human thought, paradox has been a signal that we have to look hard for explanations, whether in natural language or the symbols of math and logic. The particular resonance between math and poetry is related to the fact that paradoxical concepts can translate from one form of expression to another surprisingly well. The author examines paradoxes that have intrigued her and presents five of her poems that have been inspired as a result. Keywords: mathematical poetry; paradox; Zeno; Eubulides; Russell’s Paradox; Twin Paradox; Liar’s Paradox AMS Subject Classification: 00A99; 03A10; 03B99; 97E20 1. Russell and Zeno Paradox has long been a mental sandbox in which mathematicians, logicians and poets like to play around. A paradox is something that is both true and not true, simultaneously logical and illogical. Its contradictions can seem trivial, nonsensical, and yet lead on to powerful insight. And it serves as a hinge between two modes of thought—mathematical and artistic—that are often considered different. Perhaps this is the case because, at heart, mathematics and art are deeply paradoxical activities. “The paradox of the arts is that they are all made up and yet they allow us to get at truths about who and what we are or might be…” wrote poet Seamus Heaney [10, p 69]. -
Philosophy Sunday, July 8, 2018 12:01 PM
Philosophy Sunday, July 8, 2018 12:01 PM Western Pre-Socratics Fanon Heraclitus- Greek 535-475 Bayle Panta rhei Marshall Mcluhan • "Everything flows" Roman Jakobson • "No man ever steps in the same river twice" Saussure • Doctrine of flux Butler Logos Harris • "Reason" or "Argument" • "All entities come to be in accordance with the Logos" Dike eris • "Strife is justice" • Oppositional process of dissolving and generating known as strife "The Obscure" and "The Weeping Philosopher" "The path up and down are one and the same" • Theory about unity of opposites • Bow and lyre Native of Ephesus "Follow the common" "Character is fate" "Lighting steers the universe" Neitzshce said he was "eternally right" for "declaring that Being was an empty illusion" and embracing "becoming" Subject of Heideggar and Eugen Fink's lecture Fire was the origin of everything Influenced the Stoics Protagoras- Greek 490-420 BCE Most influential of the Sophists • Derided by Plato and Socrates for being mere rhetoricians "Man is the measure of all things" • Found many things to be unknowable • What is true for one person is not for another Could "make the worse case better" • Focused on persuasiveness of an argument Names a Socratic dialogue about whether virtue can be taught Pythagoras of Samos- Greek 570-495 BCE Metempsychosis • "Transmigration of souls" • Every soul is immortal and upon death enters a new body Pythagorean Theorem Pythagorean Tuning • System of musical tuning where frequency rations are on intervals based on ration 3:2 • "Pure" perfect fifth • Inspired -
Paradoxes Situations That Seems to Defy Intuition
Paradoxes Situations that seems to defy intuition PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 08 Jul 2014 07:26:17 UTC Contents Articles Introduction 1 Paradox 1 List of paradoxes 4 Paradoxical laughter 16 Decision theory 17 Abilene paradox 17 Chainstore paradox 19 Exchange paradox 22 Kavka's toxin puzzle 34 Necktie paradox 36 Economy 38 Allais paradox 38 Arrow's impossibility theorem 41 Bertrand paradox 52 Demographic-economic paradox 53 Dollar auction 56 Downs–Thomson paradox 57 Easterlin paradox 58 Ellsberg paradox 59 Green paradox 62 Icarus paradox 65 Jevons paradox 65 Leontief paradox 70 Lucas paradox 71 Metzler paradox 72 Paradox of thrift 73 Paradox of value 77 Productivity paradox 80 St. Petersburg paradox 85 Logic 92 All horses are the same color 92 Barbershop paradox 93 Carroll's paradox 96 Crocodile Dilemma 97 Drinker paradox 98 Infinite regress 101 Lottery paradox 102 Paradoxes of material implication 104 Raven paradox 107 Unexpected hanging paradox 119 What the Tortoise Said to Achilles 123 Mathematics 127 Accuracy paradox 127 Apportionment paradox 129 Banach–Tarski paradox 131 Berkson's paradox 139 Bertrand's box paradox 141 Bertrand paradox 146 Birthday problem 149 Borel–Kolmogorov paradox 163 Boy or Girl paradox 166 Burali-Forti paradox 172 Cantor's paradox 173 Coastline paradox 174 Cramer's paradox 178 Elevator paradox 179 False positive paradox 181 Gabriel's Horn 184 Galileo's paradox 187 Gambler's fallacy 188 Gödel's incompleteness theorems -
The Π Research Network
Version 1.0 THE Π RESEARCH NETWORK Patrick Allo, Bert Baumgaertner, Simon D’Alfonso, Nir Fresco, Federico Gobbo, Carson Grubaugh, Andrew Iliadis, Phyllis Illari, Eric Kerr, Giuseppe Primiero, Federica Russo, Christoph Schulz, Mariarosaria Taddeo, Matteo Turilli, Orlin Vakarelov, Hector Zenil. THE PHILOSOPHY OF INFORMATION AN INTRODUCTION T H E Π RESEARCH NETWORK The Philosophy of Information An Introduction The Philosophy of Information - An Introduction by The Π Research Network is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. THE PHILOSOPHY OF IN F O R M A T I O N — A N INTRODUCTION Table of Contents Table of Contents 1 List of Figures 5 PREFACE 6 CONTRIBUTORS 7 Part I: Introductory material 8 1. A QUICK HISTORY OF THE PHILOSOPHY OF INFORMATION 9 1.1 Introduction 9 1.2 Turing’s basic idea 10 1.3 Shannon’s basic idea 12 1.4 Extension of the concepts 14 1.5 Cybernetics 15 1.6 Dretske 18 1.7 French Philosophy of Information 21 1.8 Conclusion 26 1.9 Exercises 26 1.10 Further reading 27 2. WHAT IS THE PHILOSOPHY OF INFORMATION TODAY? 28 2.1 Introduction 28 2.2 The information revolution alters our self-understanding 29 2.3 The philosophy of information as a field 31 2.4 Open and closed questions 33 2.5 The idea of a Level of Abstraction (LoA) 36 2.6 The definition of a level of abstraction 37 2.7 The implications of LoAs 39 2.8 Exercises 41 2.9 Further reading 42 3. NATURALISED INFORMATION 43 3.1 Semantic vs. -
List of Paradoxes 1 List of Paradoxes
List of paradoxes 1 List of paradoxes This is a list of paradoxes, grouped thematically. The grouping is approximate: Paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only those instances that have been termed paradox by at least one source and which have their own article. Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis. Logic • Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. • What the Tortoise Said to Achilles "Whatever Logic is good enough to tell me is worth writing down...," also known as Carroll's paradox, not to be confused with the physical paradox of the same name. • Crocodile Dilemma: If a crocodile steals a child and promises its return if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned? • Catch-22 (logic): In need of something which can only be had by not being in need of it. • Drinker paradox: In any pub there is a customer such that, if he or she drinks, everybody in the pub drinks. • Paradox of entailment: Inconsistent premises always make an argument valid. • Horse paradox: All horses are the same color. • Lottery paradox: There is one winning ticket in a large lottery. It is reasonable to believe of a particular lottery ticket that it is not the winning ticket, since the probability that it is the winner is so very small, but it is not reasonable to believe that no lottery ticket will win. -
The Barber Paradox: on Its Paradoxicality and Its Relationship to Russell’S Paradox
Prolegomena 13 (2) 2014: 269–278 The Barber Paradox: On its Paradoxicality and its Relationship to Russell’s Paradox JIRI RACLAVSKY Department of Philosophy, Masaryk University, Arne Novaka 1, 602 00 Brno, Czech Republic [email protected] ORIGINAL SCIENTIFIC ARTICLE / RECEIVED: 210914 ACCEPTED: 261014 ABSTRACT: The Barber Paradox is often introduced as a popular version of Russell’s paradox, though some philosophers and logicians (e.g. Church) have denied their similarity, even calling the Barber paradox a pseudoparadox. In the first part of the paper, we demonstrate mainly that in the standard (Quinean) definition of a paradox the Barber paradox is a clear-cut example of a non-paradox. In the second part of the paper, we examine a probable source of the paradoxicality of the Barber Paradox, which is found in a certain ambivalence in terms of meaning. The two different read- ings of the crucial phrase yield distinct existential assumptions which produce the paradoxical conclusion. KEYWORDS: Degrees of paradoxicality, pseudoparadoxicality, Russell’s paradox, the Barber paradox. 1. Introduction One of the paradoxical aspects of the Barber paradox (BP) is that it is not a paradox, though many people still think it is.1 It is also paradoxical that its authorship is often attributed to Russell, 2 even though he did not invent it,3 and even warned against it as a false analogy to Russell’s paradox (RP).4 1 Including some philosophers that address paradoxes, e.g., Rescher (2001: 143–147). 2 Cf., e.g., the popular text Joyce (2002). 3 The probable author of the BP is the Austrian philosopher Mally. -
What Is Mathematics: Gödel's Theorem and Around. by Karlis
1 Version released: January 25, 2015 What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book "Around Gödel's theorem" published in 1992 in Russian (online copy). Diploma, 2000 Diploma, 1999 This work is licensed under a Creative Commons License and is copyrighted © 1997-2015 by me, Karlis Podnieks. This hyper-textbook contains many links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. Are you a platonist? Test yourself. Tuesday, August 26, 1930: Chronology of a turning point in the human intellectua l history... Visiting Gödel in Vienna... An explanation of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). 2 Table of Contents References..........................................................................................................4 1. Platonism, intuition and the nature of mathematics.......................................6 1.1. Platonism – the Philosophy of Working Mathematicians.......................6 1.2. Investigation of Stable Self-contained Models – the True Nature of the Mathematical Method..................................................................................15 1.3. Intuition and Axioms............................................................................20 1.4. Formal Theories....................................................................................27 -
Critique of Previous Comprehensive Studies of Self-Referential Paradoxes
Critique of Previous Comprehensive Studies of Self-referential Paradoxes YAMAMORI Maiko Kyoto University Abstract: The fact that it produces a paradox implies the possibility that a theory or a thought is not entirely rational. At the same time, the possible resolution of paradoxes suggests candidates for the conditions that may make theories or thoughts consistent. Many scholars have investigated paradoxes to find their solutions. The primary approach to a paradox is to consider each solution in turn (the one-by-one approach); studies using this approach have resulted in many diverse solutions, but it is limited by serious problems. Hence, I will adopt another approach, termed here as the comprehensive approach, to consider a general solution or solutions to all paradoxes. Here the focus will be on self-referential paradoxes. A self-referential paradox indicates a paradox that occurs when the subject of a proposition is in part the proposition itself. Further, some paradoxes are not self-referential but seem to be. To preserve the appearances of our intuition of a self-reference, I introduce a categorization of self-references into two types (with two corresponding types of accompanying paradoxes): the first is a narrow self-reference, that is, precisely a reference to itself (this matches the existing characterization). The other type is a broad self-reference or a reference to a group or groups containing itself. In this paper, first, I will examine the works of three figures, Bertrand Russell; Graham Priest; and Martin Pleitz, who take a comprehensive approach: considering the general conditions for self-referential paradoxes, analyzing them, and suggesting general solutions for them. -
Mathematical Paradoxes
Introduction: Mathematical Paradoxes Intuitive approach. Until recently, till the end of the 19th century, mathematical theories used to be built in an intuitive or axiomatic way. The historical development of mathematics has shown that it is not sufficient to base theories on an intuitive understanding of their notions only. This fact became especially obvious in set theory. 1 The basic concept of a set is certainly taken from reality, for there we come across many examples of various sets, all of which are finite. But in mathematics it is also necessary to consider infinite sets, such as the set of all integers, the set of all rational numbers, the set of all segments, the set of all tri- angles. 2 Intuitively, by a set, we mean any collection of objects- for example, the set of all even integers or the set of all students in a class. The objects that make up a set are called its members (elements). Sets may themselves be members of sets for example, the set of all sets of integers has sets as its members. 3 Most sets are not members of themselves. For example the set of all students is not a member of itself, because the set of all students is not a student. However, there may be sets that do belong to themselves. For example the set of all sets. A following simple reasoning indicates that it is necessary to impose some limitations on the concept of a set. 4 Russell Paradox, 1902: Consider the set A of all those sets X such that X is not a member of X. -
Basics of Set Theory
2.2. THE BASICS OF SET THEORY In this section we explain the importance of set theory and introduce its basic concepts and definitions. We also show the notation with sets that has become standard throughout mathematics. The Importance of Set Theory One striking feature of humans is their inherent propensity, and ability, to group objects according to specific criteria. Our prehistoric ancestors were hunter-gatherers who grouped tools based on their survival needs. They eventually settled and formed strict hierarchical societies where a person belonged to one class and not another. Today, many of us like to sort our clothes at home or group the songs on our computer into playlists. As we look at the accomplishments of science at the dawn of this new millennium, we can point to many impressive classifications. In chemistry there is Mendeleev’s periodic table, which lists all the known chemical elements in our universe and groups them based on common structural characteristics (alkaline metals, noble gases, etc.) In biology there is now a vast taxonomy that systematically sorts all living organisms into specific hierarchies: kingdoms, orders, phyla, genera, etc. In physics, all the subatomic particles and the four fundamental forces of nature have now been classified under an incredibly complex and refined theory called the Standard Model. This model – the most accurate scientific theory ever devised – surely ranks as one of mankind’s greatest achievements. The idea of sorting out certain objects into similar groupings, or sets, is the most fundamental concept in modern mathematics. The theory of sets has, in fact, been the unifying framework for all mathematics since the German mathematician Georg Cantor formulated it in the 1870’s. -
Diagonalization and Logical Paradoxes
DIAGONALIZATION AND LOGICAL PARADOXES DIAGONALIZATION AND LOGICAL PARADOXES By HAIXIA ZHONG, B.B.A., M.A. A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy McMaster University © Copyright by Haixia Zhong, August 2013 DOCTOR OF PHILOSOPHY (2013) (Philosophy) McMaster University Hamilton, Ontario TITLE: Diagonalization and Logical Paradoxes AUTHOR: Haixia Zhong, B.B.A. (Nanjing University), M.A. (Peking University) SUPERVISOR: Professor Richard T. W. Arthur NUMBER OF PAGES: vi, 229 ii ABSTRACT The purpose of this dissertation is to provide a proper treatment for two groups of logical paradoxes: semantic paradoxes and set-theoretic paradoxes. My main thesis is that the two different groups of paradoxes need different kinds of solution. Based on the analysis of the diagonal method and truth-gap theory, I propose a functional-deflationary interpretation for semantic notions such as ‘heterological’, ‘true’, ‘denote’, and ‘define’, and argue that the contradictions in semantic paradoxes are due to a misunderstanding of the non-representational nature of these semantic notions. Thus, they all can be solved by clarifying the relevant confusion: the liar sentence and the heterological sentence do not have truth values, and phrases generating paradoxes of definability (such as that in Berry’s paradox) do not denote an object. I also argue against three other leading approaches to the semantic paradoxes: the Tarskian hierarchy, contextualism, and the paraconsistent approach. I show that they fail to meet one or more criteria for a satisfactory solution to the semantic paradoxes. For the set-theoretic paradoxes, I argue that the criterion for a successful solution in the realm of set theory is mathematical usefulness.