DOCSLIB.ORG

Ebook Download Paradoxes in Mathematics Ebook, Epub

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ebook Download Paradoxes in Mathematics Ebook, Epub PARADOXES IN MATHEMATICS PDF, EPUB, EBOOK Stanley J. Farlow | 192 pages | 16 Apr 2014 | Dover Publications Inc. | 9780486497167 | English | New York, United States Paradoxes in Mathematics PDF Book Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. Reinhardt, W. The phrase 'one does not shave oneself' also includes the following cases along with the usual case in which one is shaved by another person: 1. When Achilles reaches T 1 , the labouring Tortoise will have moved on 0. It can feature any verb or verb phrase - shave, adore, paint, include in a set, share a pizza with - that can take a subject, object, and crucially, a tense. Teaching Area. True So let's study the action of creating that stone which God cannot lift up. Betti, A. For instance, there is no precise criterion for deciding whether a given expression of the natural language represents a rule uniquely defining a number. For example, The Banach-Tarski paradox can be considered such a paradox. You can quibble about what precisely is meant by the term "paradox," but if I understand your meaning, the so-called liar paradoxes can be seen as logical paradoxes. The role of uniformity is essential in previous investigations. If one is given free rein to do anything at all in set theory, a set could contain itself. McGee, V. Remove all the hair from your face? The father of set theory, Cantor, had noticed similar difficulties already in as witnessed by Bernstein and by letters to Hilbert and Dedekind. It contains a new paradox credited to Grelling with a semantical flavor see also the entry self- reference :. Other editions. Bertrand Russell's discovery of this paradox in dealt a blow to one of his fellow mathematicians. On the positive side, the concept of truth can be adequately defined for any formalized language L in a language the so-called metalanguage , provided it is of higher order than L. Same goes for "shave". That proves that the paradox does not take place. Dominic James marked it as to-read Aug 31, Therefore, paradoxes must be banished. Remove "finish line" and "time". Mancosu, P. Some terms are formally provable or assertable and are classified as true. Dean, W. I used your Barber paradox in my blog today and included a link to the page I used. Between the end of the 19th century and the beginning of the 20th century, the foundations of logic and mathematics were affected by the discovery of a number of difficulties—the so-called paradoxes—involving fundamental notions and basic methods of definition and inference , which were usually accepted as unproblematic. Paradoxes in Mathematics Writer This happens when we have a set with two independent variables, which means they should be entirely unrelated. Types are intrinsic to logical and mathematical objects and the logical paradoxes are exactly those which require type distinctions to be solved e. The essential point is that each propositional function has a range of significance, i. Besides the axiom of infinity, AR is an essential tool for reconstructing classical mathematics, but it is a strong existential principle, apparently in conflict with the philosophical idea that logical and mathematical entities are to be constructively generated according to the vicious circle principle. Farlow , a prolific math writer with ten other, more advanced titles according to his Goodreads credits, does a very good job explaining mathematical concepts in the simplest manner the material allows. But this notion was not so easy to accept. These are sometimes written on two sides of a card "The statement on the other side is true. But why would this problem be worth studying from a logical and mathematical point of view? Those men that do not shave themselves are shaved by the barber. Permalink Submitted by Anonymous on February 6, The original liar paradox, for example, is based on a claim by the Cretan poet, Epimenides circa BC , that Cretans are "always liars. It is confusing. His logical analysis leads to the conclusion that the paradoxes involve meaningless statements. If one looks closely at the development of these systems, one can see that paradoxical constructions have become essential tools for defining objects and proving non- trivial logical mathematical facts. Phenomenological paradoxes are banished by either fine-tuning the axioms so that the paradoxical result no longer follows or by accepting the result as true and announce that our intuition has been refined. Anderson, C. Sun Wukong actually peed on Buddha's hand and wrote insults on it. Yes, but he does not barber himself. I shave anyone who does not shave himself, and noone else" it means that he is a barber who obviously works in his shop during working hours, i. This makes his theory quite expressive e. Theories of naive truth—as based on the unrestricted biconditional and on a logic without contraction—are to be found in the literature, e. Barwise, J. He no longer insisted on the vicious circularity of the definition involved in the contradictions; instead, he held the view that a predicative classification is characterized by invariance , i. Unless explicitly noted otherwise, all content licensed as indicated by RationalWiki:Copyrights. Tarski, A. Hessenberg, G. The parameters of the question are undefined. Church, A. Friedman, H. So the contradiction is ascribed to an error in the theory of definitions, namely to the use of definitions that give rise to an infinite chain of substitutions, without converging to a result. John L. Now it is a little clearer. Paradoxes in Mathematics Reviews This avoids the possibility of having to talk about the set of all sets that are not members of themselves, because the two parts of the sentence are of different types - that is, at different levels. Russell's Paradox, for example, would be more rightly called Russell's Contradiction in my terminology. Richard, J. One might think that the development of logic and set theory in the 20th century has exorcized paradoxes, and that contradictions in logical systems is a phenomenon of the years of foundational crisis only. In fact even Zeno's belief in monism - in a static, unchanging reality - which was the basis for his producing the arguments in the first place, seems oddly similar to cosmologists ideas about ' worldlines ' the 'history' of a particle in spacetime where 'the entire history of each worldline already exists as a completed entity in the plenum of space time' read more. The role of contraction was noticed by Fitch , who observed that, in order to derive the Russell paradox one considers a function of two variables, then one diagonalizes and regards such an object as a new unary propositional function. While these are powerful resources, they also open the door to people without a full statistical understanding to misunderstand some of the subtleties within a dataset and to draw wildly incorrect conclusions. Remove "finish line" and "time". Most mathematicians are interested primarily in distinguishing the true from the false, and do not concern themselves with things that have no truth value. For example, if patients only present at a clinic with disease A, disease B or both, then even if the two diseases are independent, a negative association between them may be observed. Since the year , this research thread has been intensively studied with various aims, from proof theoretic analysis to philosophical discussion of minimalism for a survey of the varieties of truth theoretic systems and appropriate references, see the entry on axiomatic theories of truth and the recent monographs of Halbach , Horsten ; see also the papers Feferman , Fujimoto , and , Horsten and Leigh forthcoming, Leigh and Rathjen and , Leigh , a, b, Enayat and Visser Church, A. Given this semantical machinery, Tarski can solve in the negative the problem of the existence of a formal counterpart of a universal language, i. B, 13 , No. You might call that the Madison Avenue theory, or the P. In Oppositions and Paradoxes , John L. Author Stanley J. What is a paradox in mathematics? Now what about the set of all sets which are not members of themselves? In one of the first books of Journey to the West, Sun Wukong is tasked with leaving the palm of the hand of buddha. Boffa and D. Any clearly phrased condition is thought to define a set - namely, those things that satisfy the condition. Space and time are one thing. How ever, I do straight razor shave men. Rendiconti del Circolo Matematico di Palermo , , — English translation in van Heijenoort , — Truth and membership are inductively generated by iterating rules that correspond to natural logical closure conditions and can be formalized by means of positive i. Wood eds. Helen Joyce. Tarski eds. Paradoxes in Mathematics Read Online B, 13 , No. For example Burali-Forti shows that there is no set which contains "all the ordinals". This is especially true for the notions of set and collection in general, for the basic syntactical and semantical concepts of standard classical logic logical languages of a given order, the notion of satisfiability, definability. It is different aspects of self. People often disbelieve this, recalling that it is rare that they meet someone who shares their own birthday. But impredicativity makes the construction of a model or of an interpretation more difficult and less evident. Dissertation, Amsterdam English translation in L. Now, the lamp is initially off and I switch it on. Though most sets don't. Yours Mysteriously, Anonymous. He also states that it is possible to display the new computable continuum in a hierarchy i. Tarski, Logic, Semantics, Metamathematics , 2d ed.
Load more

Recommended publications
  • 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].
    [Show full text]
  • 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
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Paradoxes in Logic and Set Theory
    MATH110 Lecturenotes–11 Paradoxes in Logic and Set Theory 1. Existence of irrational numbers. About 2.5 thousand years ago, Pythagoreans proved that √2 was irrational, which was controversial at the time. Until then it was believed that all numbers are rational, i.e. can be expressed as quotients of integers. 2. Paradox of enumeration. Which set is bigger? Before formal set theory was introduced, the notion of the size of a set had been problematic. It had been discussed by Italian mathematicians Galileo Galilei and Bernard Bolzano (17th-18th centruries), among others. Are there more as as many natural numbers as perfect squares (squares of natural numbers)? The answer is yes, because for every natural number n there is a square number n2, and likewise the other way around. The answer is no, because the squares are a proper subset of the naturals: every square is a natural number but there are natural numbers, like 2, which are not squares of natural numbers. By defining the notion of the size of a set in terms of its cardinality, the issue can be settled, as there is a bijection between the two sets involved. Georg Cantor (late 19th-early 20th centuries) invented set theory. Cantor es- tablished the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real num- bers are more numerous than the natural numbers. He introduced the concept of cardinality. “I see it but I don’t believe,” Cantor wrote to Dedekind after proving that the set of points of a square has the same cardinality as that of the points on just an edge of the square: the cardinality of the continuum.
    [Show full text]
  • Mathematics' Quirks: the Barber Paradox
    The Clovis Tutorial Center’s I N S I D E THIS ISSUE: WRITEVOLUME 16, ISSUE 4 NOWDECEMBER 2016 Leadership 2 Stories and 2 Legends Mathematics’ Quirks: The Barber Paradox Joining the By Taylor Lopez 3 Conversation I love paradoxes; they make ry this problem can not be that couldn’t create a loop Essential Math 3 people question the way resolved. This paradox was like Russell’s paradox. After Pick-Up Lines that humans use some theo- discovered in 1901 by the publishing the three vol- Tutors of the ries to interpret the world. aforementioned Bertrand umes of Principia Mathemat- 4 Month Can things we take for com- Russell. At the time, this ica in 1910, 1912, 1913, and mon knowledge be incor- was a huge problem for the a second edition in 1927, his Working at rect? Does 1+1 really equal foundations of mathematics. work was highly regarded as 4 the Zoo 2? One of my favorite phi- One hundred years previ- the greatest logical theory losophers, Bertrand Russell, ous, Gottlob Frege tried to since Aristotle’s Organon. Sudoku 5 attempted to solve a para- develop a foundation for all Unfortunately Kurt Gödel’s dox of set theory in his mathematics using symbolic incompleteness theorem Word of the magnum opus Principia Math- logic. Frege established con- that finally showed that the 5 Month ematica. Russell’s paradox can nections like x=5, stating Principia Mathematica could be described with a riddle, “If the the value of x is 5. This not be both consistent and Math It! 5 a barber shaves all and only makes intuitive sense but complete.
    [Show full text]
  • [Uncommon Sense: the Sensual, Acoustic Logic [S] of Paradox and Art
    Sheena Calvert: [Unjdisciplined Gestures and [Unjcommon Sense [Un]disciplined Gestures and [Uncommon sense: The Sensual, Acoustic Logic [s] of Paradox and Art Sheena Calvert This thesis is submitted as a partial requirement towards a Doctorate of Philosophy at the University of Greenwich December, 2009 Supervisory Committee Prof. Johnny Golding Prof John Williams Sheena Calvert: [Unjdisciplined Gestures and [Unjcommon Sense Acknowledgements I would like to thank my mother, Edna Calvert, my father, John Calvert, my Aunt, Marion Wilkes, my family in Cleveland, Steven Popper, and my 'extended family' of loyal and caring friends, including the University of Greenwich classmates, who are a constant source of inspiration for the work. My very special thanks to Bex Muirhead, who has taken 'loyalty' and love to a new level. Each in their own ways has contributed to the work, in terms of invaluable intellectual, emotional, and practical support. The late Mark Whistler taught me many things, including how to 'hear' Twombly, and I miss his unique ability to make the mundane magnificent. Johnny Golding has consistently reminded me throughout this process, of the need to resist 'looking down or back'. Without her intense intellectual and personal guidance, the work simply would not exist. John Williams has provided many generous insights and comments on the text. My sincere thanks to both. An AHRC Doctoral Award funded the first three years of this research and I am endebted to them for that support. Finally, this thesis is dedicated to my parents, John and Edna Calvert, both of who taught me the value of 'experience', and to Flossie and Bb, who brought me so much simple joy along the way.
    [Show full text]
  • The Barber Paradox: on Its Paradoxicality and Its Relationship to Russell’S Paradox
    The Barber Paradox: on its Paradoxicality and its Relationship to Russell's Paradox Jiˇr´ıRaclavsk´y Abstract The Barber paradox is often introduced as a popular version of Rus- sell's paradox, though some experts have denied their similarity, even calling the Barber paradox a pseudoparadox. In the first part of the paper, I demonstrate mainly that in the standard (Quinean) defini- tion of a paradox the Barber paradox is a clear-cut example of a non- paradox. Despite some outward similarities, it differs radically from Russell's paradox. I also expose many other differences. In the second part of the paper, I 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 readings of the crucial phrase yield distinct existential assumptions which produce the paradoxical conclusion. One of the paradoxical aspects of the Barber paradox (BP) is that it is not a paradox, though many people still think it is (including some philosophers that address paradoxes, e.g., Nicholas Rescher, [14], pp. 143−147). It is also paradoxical that its authorship is often attributed to Bertrand Russell (e.g. [8]), even though he did not invent it,1 and even warned against it as a false analogy to Russell's paradox (RP).2 Finally, it is also somewhat paradoxical that despite what I have previously written, I will suggest a way to construe the BP as a real paradox. This paper begins with an exposition of both RP and the BP, mainly showing the ways in which they are not similar.
    [Show full text]
  • Cesare Burali-Forti
    RAMACHANDRAN P.T. RETIRED PROFESSOR & HEAD OF MATHEMATICS, UNIVERSITY OF CALICUT E-mail: [email protected] Phone: 0483-2754300 Mob: 9447791504 Website: www.ramachandranpt.in PHILOSOPHY OF MATHEMATICS Lecture 3 LAW OF EXCLUDED MIDDLE Every statement is either true or false but not both. MANY VALUED LOGICS In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. Classical two-valued logic may be extended to n-valued logic for n greater than 2. Those most popular in the literature are three- valued (e.g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many valued), such as fuzzy logic and probability logic. IGNORANCE AND KNOWLEDGE Without ignorance, there would be no knowledge, without knowledge, there would be no ignorance. SET THEORIES • Fuzzy Set Theory • Rough Set Theory • Soft Set Theory • Multi Sets • Nonstandard Set Theory • Alternative Set Theory PEANO’S AXIOMS 1. 0 is a number 2. The successor of any number is a number 3. No two numbers have the same successor 4. 0 is not the successor of any number 5. Any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers ( Induction Axiom) INDUCTION FAILS An assertion may be valid in an entire series of particular cases, and at the same time invalid in general PRINCIPLE OF MATHEMATICAL INDUCTION P(n) for n= 1,2,3… P(1) is true P(k) implies P(k+1) for all k Then P(n) is true for all n=1,2,3… AXIOMATIC METHOD EULID • Elements (c.
    [Show full text]
  • David Keyt Two Paradoxes 1) the Barber Paradox: in an All Male Army
    David Keyt Two Paradoxes 1) The Barber Paradox: In an all male army one member of a squad of soldiers is a barber who shaves all and only those members of the squad who do not shave themselves. Who shaves the barber? 2) What It Shows: i) There is a member of the squad who shaves all and only those members of the squad who do not shave themselves. Call him ‘John’. ii) For each member of the squad, x, John shaves x iff it is false that x shaves x. iii) But John is a member of the squad. iv) So John shaves John iff it is false that John shaves John, a contradiction. v) Therefore, (i) is false: there is no such barber. 3) The Barber's Paradox Resolved: a) Let’s make a minimal change in our description of the squad of soldiers: i) There is a someone who shaves all and only those members of the squad who do not shave themselves. Let’s call this person ‘Barber’. ii) So Barber shaves all and only those members of the squad who do not shave themselves. iii) That is to say, for all x, if x is a member of the squad, then Barber shaves x iff x does not shave x. iv) Hence, if Barber is a member of the squad, then Barber shaves Barber iff Barber does not shave Barber. v) But it is false that Barber shaves Barber iff Barber does not shave Barber. vi) Therefore, by modus tollens Barber is not a member of the squad.
    [Show full text]