Goedel's Way: Exploits into an undecidable world PDF
Preview Goedel's Way: Exploits into an undecidable world
Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best C h known for his incompleteness theorems. He was the greatest mathematical a logician of the 20th century, with his contributions extending to Einstein’s itin , general relativity, as he proved that Einstein’s theory allows for time machines. d a C o The Gödel incompleteness phenomenon - the usual formal mathematical systems s ta cannot prove nor disprove all true mathematical sentences - is frequently presented , D in textbooks as something that happens in the rarefied realms of mathematical o r logic, and that has nothing to do with the real world. Practice shows the contrary ia though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics, and even ecology. In this lively treatise, based on Chaitin’s groundbreaking work and on the da Costa- Doria results in physics, ecology, economics and computer science, the authors show that the Gödel incompleteness phenomenon can directly bear on the G practice of science and perhaps on our everyday life. ö d This accessible book gives a new, detailed and elementary explanation of the e Gödel incompleteness theorems and presents the Chaitin results and their l ’ relation to the da Costa-Doria results, which are given in full, but with no s technicalities. Besides theory, the historical report and personal stories about W the main character and on this book’s writing process, make it appealing a leisure reading for those interested in mathematics, logic, physics, philosophy y and computer sciences. Gregory Chaitin is an Argentinian-American mathematician and computer scientist. The author of many books and scholarly papers, Chaitin Gödel’s Way proved the Gödel-Chaitin incompleteness theorem and is the discoverer of the remarkable Omega number, which shows that God plays dice in pure mathematics. Newton da Costa is a Brazilian logician whose best known Exploits into an contributions have been in the realms of nonclassical logics and philosophy of science. Da Costa developed paraconsistent logics, that is, logical systems that undecidable world admit inner contradictions. Francisco Antonio Doria is a Brazilian physicist. He has made contributions to the gauge field copy problem in quantum field theory and proved with Newton da Costa several incompleteness theorems in mathematics, physics and mathematical economics, including the undecidability of chaos theory. Gregory Chaitin Newton da Costa Francisco Antonio Doria an informa business G¨odel’s Way TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Apersonalaccountbysomeoftheparticipantsintheworkgoing beyondGo¨delbyfindinguncomputabilityandincompletenessin manyareasofcontinuousanddiscretemathematicsandtheoreti- calphysics. TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk G¨odel’s Way Exploits into an undecidable world Gregory Chaitin, Newton da Costa & Francisco Antonio Doria CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20120127 International Standard Book Number-13: 978-0-203-16957-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Onnerec¸oitpaslasagesse, ilfautlade´couvrirsoi-meˆme, apre`suntrajetquepersonne nepeutfairepournous, nepeutnouse´pargner, carelleest unpointdevuesurleschoses MarcelProust TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Contents Prologue xiii Acknowledgments xvii AbouttheAuthors xix ACaveat xxi 1.Go¨del,Turing 1 Go¨del:logicandtime 2 Ashortbiography 4 Theincompletenesstheorems,I 5 Kleene’sversionofthefirstincompletenesstheorem 6 AnimmediateconsequenceofKleene’sproof 7 The incompleteness theorems II: consistency cannot be proved withinthesystem 8 Aweirdformalsystem 9 Canweprovetheconsistencyofarithmetic? 10 Chaitin’sincompletenesstheorem 11 Berry’sParadox 12 Rice’stheorem 13 MoreworkbyGo¨del:theconstructiveuniverseofsets 14 Aconcludingnote:Go¨delontimemachines 17 AlanTuringandhismathematicalmachines 20 Whatisacomputation? 20 Turingmachines,I 21 Turingmachines,II 22 Theuniversalmachine 23 Thehaltingproblem 23 Go¨del’sfirstincompletenesstheoremrevisited 24 TheChurch–Turingthesis 24 Diophantineequations;Hilbert’s10thproblem 25 Undecidableissues 27 ix
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