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“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
The Repah Report Without Appendices
© Stephen Brown, Robb Ross, David Gerrard, De Montfort University Mark Greengrass, Jared Bryson, Sheffield University Published by: HriOnline for The RePAH Project Knowledge Media Design & The Humanities Research Institute De Montfort University University of Sheffield Portland 2.3a 34 Gell St The Gateway Sheffield South Yorkshire S10 2TN Leicester LE1 9BH ISBN: 0-9542608-8-0 Also Available at http://repah.dmu.ac.uk/report The right of Stephen Brown, Robb Ross, David Gerrard, Mark Greengrass and Jared Bryson to be identified as the Authors of this Work has been asserted by them in accordance with the Copyright, Designs and Patents Act of 1988. September 2006 Contents Contents .............................................................................................................................. 4 1. Acknowledgements........................................................................................................ 6 2. Executive Summary....................................................................................................... 7 2.1 The Work of the Project........................................................................................... 7 2.3 Conclusions............................................................................................................... 7 2.3.1 Users’ information discovery strategies and internet usage .............................. 8 2.3.2 Information about users’ awareness and attitudes with respect to currently available online services and tools, including such gateways and -
Unfolding of Systems of Inductive Definitions
UNFOLDINGOFSYSTEMSOFINDUCTIVE DEFINITIONS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE OF GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FORTHEDEGREEOF DOCTOROFPHILOSOPHY Ulrik Torben Buchholtz December 2013 © 2013 by Ulrik Torben Buchholtz. All Rights Reserved This work is licensed under the Creative Commons Attribution 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/us/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. This dissertation is online at: http://purl.stanford.edu/kg627pm6592 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Solomon Feferman, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Grigori Mints, Co-Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Thomas Strahm Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education An original signed hard copy of the signature page is on file in University Archives. iii iv ABSTRACT This thesis is a contribution to Solomon Feferman’s Unfolding Program which aims to provide a general method of capturing the operations on individuals and predicates (and the principles governing them) that are implicit in a formal axiomatic system based on open-ended schemata. -
Litterær Kompetanse
“Som å lese en film” Elevers lesing av elektronisk litteratur Pål Fredrik Børresen Mastergradsoppgave i språk, kultur og digital kommunikasjon ved Avdeling for lærerutdanning og naturvitenskap HØGSKOLEN I HEDMARK Våren 2011 2 Forord Jeg vil først og fremst rette en stor takk til Hans Kristian Rustad for grundig og konstruktiv veiledning. Jeg vil også takke læreren som velvillig gjennomførte undervisningsopplegget som denne studien bygger på, informantene som stilte opp til intervjuer, mine medstudenter for givende diskusjoner, og min kone for tålmodig støtte. Interesserte kan få tilgang til transkriberingene av intervjuene som er gjort i forbindelse med denne oppgaven, ved å henvende seg til meg. Pål Fredrik Børresen Hamar, mai 2011 3 Innhold Forord ....................................................................................................................................... 2 Innhold ...................................................................................................................................... 3 1. Innledning ......................................................................................................................... 6 1.1 Hva er elektronisk litteratur? ................................................................................... 10 1.2 Forskning på elektronisk litteratur ........................................................................... 14 1.3 Mål og problemstilling ............................................................................................ 23 2. Teoretisk rammeverk -
Notices of the American Mathematical Society
June 18 and 19)- Page 341 Vl 0 ~ Mathematical Society Calendar of AMS Meetings THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of the Notices was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab· stracts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. MEETING ABSTRACT NUMBER DATE PLACE DEADLINE -
Proof Theory Synthese Library
PROOF THEORY SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Boston University Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley THEO A.F. KUIPERS, University of Groningen, The Netherlands PATRICK SUPPES, Stanford University, California JAN WOLENSKI, iagiellonian University, Krakow, Poland VOLUME 292 PROOF THEORY History and Philosophical Significance Edited by VINCENT F. HENDRICKS University of Copenhagen. Denmark STIG ANDUR PEDERSEN and KLAUS FROVIN J0RGENSEN University of Roskilde, Denmark SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5553-8 ISBN 978-94-017-2796-9 (eBook) DOI 10.1007/978-94-017-2796-9 Printed on acid-free paper All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1st edition 2000 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 information storage and retrieval system, without written permission from the copyright owner. CONTENTS Preface ................................................................. ix Contributing Authors ................................................. , xi HENDRICKS, PEDERSEN AND J0RGENSEN / Introduction .......... 1 PART 1. REVIEW OF PROOF THEORY SOLOMON FEFERMAN / Highlights in Proof Theory ................ , 11 1. Review of Hilbert's Program and Finitary Proof Theory ......... , 11 2. Results of Finitary Proof Theory via Gentzen's L-Calculi ......... 14 3. Shifting Paradigms .............................................. 21 4. Countably Infinitary Methods (Getting the Most Out of Logic) .. -
An Interview with Martin Davis
Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Geometry Ramanujan‘s Lost Notebook An Introduction to Mathematical of the American Mathematical Society Selected Topics in Plane and Solid Part II Cryptography May 2008 Volume 55, Number 5 Geometry G. E. Andrews, Penn State University, University J. Hoffstein, J. Pipher, J. Silverman, Brown J. Aarts, Delft University of Technology, Park, PA, USA; B. C. Berndt, University of Illinois University, Providence, RI, USA Mediamatics, The Netherlands at Urbana, IL, USA This self-contained introduction to modern This is a book on Euclidean geometry that covers The “lost notebook” contains considerable cryptography emphasizes the mathematics the standard material in a completely new way, material on mock theta functions—undoubtedly behind the theory of public key cryptosystems while also introducing a number of new topics emanating from the last year of Ramanujan’s life. and digital signature schemes. The book focuses Interview with Martin Davis that would be suitable as a junior-senior level It should be emphasized that the material on on these key topics while developing the undergraduate textbook. The author does not mock theta functions is perhaps Ramanujan’s mathematical tools needed for the construction page 560 begin in the traditional manner with abstract deepest work more than half of the material in and security analysis of diverse cryptosystems. geometric axioms. Instead, he assumes the real the book is on q- series, including mock theta Only basic linear algebra is required of the numbers, and begins his treatment by functions; the remaining part deals with theta reader; techniques from algebra, number theory, introducing such modern concepts as a metric function identities, modular equations, and probability are introduced and developed as space, vector space notation, and groups, and incomplete elliptic integrals of the first kind and required. -
PREFACE Does God Have a Big Toe? (Pages 1 - 13)
PREFACE Does God have a Big TOE? We are called to be in the world. Not of it. But not out of it, either.1 This book holds the mirror to a single paradox: We must live the historical moment we are in without letting that moment explain us. Christianity must bring to every culture an indigenous faith that is true to its heritage without Christianity’s becoming a culture faith. If every book has one idea, this is it for the one you are reading. The cardinal sin for modernity has been antiintellectualism, a dissenting spirit against celebrating the powers of the mind that has gripped large segments of the Christian church. The sin governing much of the church today may be ahistoricalism or antihistoricalism. Large segments of the Christian community, finding the intellectual and social company of modernism more congenial than that of postmodernism, are in retreat, having disconnected from the unprecedented, restructuring changes taking place all around them. The Christian mind is failing to comprehend the times, our times. The New Light apologetic chronicled in this book is devoted to enfranchising and energizing Christians to connect their faith with the indigenous historical place in which God has chosen them to live. It aims to jolt Christians into a sense of their own time, out of their fashionable out-of-itness. For the God who exists beyond time is the God who lives, moves, and has being in this time. The ultimate hospitality is, then, an entertainment of divine mystery in human life. Benedictine/biblical scholar Demetrius R. -
Plausibility and Probability in Deductive Reasoning
Plausibility and probability in deductive reasoning Andrew MacFie∗ Abstract We consider the problem of rational uncertainty about unproven mathe- matical statements, remarked on by Gödel and others. Using Bayesian-inspired arguments we build a normative model of fair bets under deductive uncertainty which draws from both probability and the theory of algorithms. We comment on connections to Zeilberger’s notion of “semi-rigorous proofs”, particularly that inherent subjectivity would be present. We also discuss a financial view with models of arbitrage where traders have limited computational resources. Contents 1 A natural problem: Quantified deductive uncertainty 2 1.1 Thephenomenon ......................... 2 1.2 The(modeling)problem . .. .. 3 2 Formal representation of plausibilities 4 2.1 Plausibility functions . 4 2.2 Languages............................. 5 2.3 Epistemic quality vs. computation costs . 5 2.4 Conditional plausibility . 7 3 Rational plausibilities in mathematics 7 3.1 Scoringrules............................. 7 3.2 Foundationsof“semi-rigorous proofs” . 8 arXiv:1708.09032v6 [cs.AI] 16 Dec 2019 4 Arbitrage pricing in markets with computational constraints 10 5 Conclusion 11 References 11 ∗School of Mathematics and Statistics, Carleton University. This work was done in part while the author was a visitor at the Harvard John A. Paulson School of Engineering and Applied Sciences. 1 1 A natural problem: Quantified deductive uncertainty 1.1 The phenomenon Epistemic uncertainty is usually defined as uncertainty among physical states due to lack of data or information. By information we mean facts which we observe from the external world. For example, whether it rains tomorrow is a piece of information we have not observed, so we are uncertain about its truth value. -
Defining Digital Humanities
Defining Digital Humanities © Copyrighted Material ashgate.com ashgate.com ashgate.com ashgate.com ashgate.com ashgate.com © Copyrighted Material © Copyrighted Material Anthony, Edward and Fergusson, Clara and Joey, For Wonne and Senne and ashgate.com ashgate.com ashgate.com ashgate.com ashgate.com ashgate.com © Copyrighted Material © Copyrighted Material Defining Digital Humanities A Reader Edited by MelissA TeRRAs ashgate.com JuliAnne nyHAn eDwARD VAnHouTTe ashgate.com ashgate.com hgate.com as ashgate.com ashgate.com © Copyrighted Material © Copyrighted Material © Melissa Terras, Julianne nyhan, edward Vanhoutte and all individual authors 2013 All rights reserved. no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Melissa Terras, Julianne nyhan and edward Vanhoutte have asserted their right under the Copyright, Designs and Patents Act, 1988, to be identified as the editors of this work. Published by Ashgate Publishing limited Ashgate Publishing Company wey Court east 110 Cherry street union Road Suite 3-1 Farnham Burlington, VT 05401-3818ashgate.com surrey, Gu9 7PT USA england www.ashgate.com British Library Cataloguing in Publication Data A catalogue record for this book is available from the British library The Library of Congress has cataloged the printed edition as follows: Defining digital humanities : a reader / [edited] by Melissa Terras, Julianne nyhan, and edward Vanhoutte. pages cm includes bibliographical references and index. isBn 978-1-4094-6962-9 (hardback)ashgate.com – isBn 978-1-4094-6963-6ashgate.com (pbk) – isBn 978-1-4094-6964-3 (epub) 1. -
Theses for Computation and Recursion on Concrete and Abstract Structures
Theses for Computation and Recursion on Concrete and Abstract Structures Solomon Feferman Abstract: The main aim of this article is to examine proposed theses for computation and recursion on concrete and abstract structures. What is generally referred to as Church’s Thesis or the Church-Turing Thesis (abbreviated CT here) must be restricted to concrete structures whose objects are finite symbolic configurations of one sort or another. Informal and principled arguments for CT on concrete structures are reviewed. Next, it is argued that proposed generalizations of notions of computation to abstract structures must be considered instead under the general notion of algorithm. However, there is no clear general thesis in sight for that comparable to CT, though there are certain wide classes of algorithms for which plausible theses can be stated. The article concludes with a proposed thesis RT for recursion on abstract structures. 1. Introduction. The concepts of recursion and computation were closely intertwined from the beginning of the efforts early in the 1930s to obtain a conceptual analysis of the informal notion of effective calculability. I provide a review of those efforts in sec. 2 as background to the remainder of this article, but I have nothing new to add here to the extensive historical and analytical literature.1 It is generally agreed that the conceptual analysis of effective calculability was first provided most convincingly by Turing (1936- 7). Not long before that Church (1936) had proposed identifying effective calculability with the Herbrand-Gödel notion of general recursiveness, soon enough proved equivalent to Turing computability among other suggested explications. -
Representations in Measure Theory: Between a Non-Computable Rock
REPRESENTATIONS IN MEASURE THEORY: BETWEEN A NON-COMPUTABLE ROCK AND A HARD TO PROVE PLACE DAG NORMANN AND SAM SANDERS Abstract. The development of measure theory in ‘computational’ frame- works proceeds by studying the computational properties of countable approx- imations of measurable objects. At the most basic level, these representations are provided by Littlewood’s three principles, and the associated approxima- tion theorems due to e.g. Lusin and Egorov. In light of this fundamental role, it is then a natural question how hard it is to prove the aforementioned theorems (in the sense of the Reverse Mathematics program), and how hard it is to com- pute the countable approximations therein (in the sense of Kleene’s schemes S1-S9). The answer to both questions is ‘extremely hard’, as follows: one one hand, proofs of these approximation theorems require weak compactness, the measure-theoretical principle underlying e.g. Vitali’s covering theorem. In terms of the usual scale of comprehension axioms, weak compactness is only provable using full second-order arithmetic. On the other hand, computing the associated approximations requires a weak fan functional Λ, which is a realiser for weak compactness and is only computable from (a certain comprehension functional for) full second-order arithmetic. Despite this observed hardness, we show that weak compactness, and certain weak fan functionals, behave much better than (Heine-Borel) compactness and the associated class of realisers, called special fan functionals Θ. In particular, we show that the combina- tion of any Λ-functional and the Suslin functional has no more computational power than the latter functional alone, in contrast to Θ-functionals.