Undecidability of First-Order Logic
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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 Metamathematics of Putnam's Model-Theoretic Arguments
The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
FOUNDATIONS of RECURSIVE MODEL THEORY Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive
Annals of Mathematical Logic 13 (1978) 45-72 © North-H011and Publishing Company FOUNDATIONS OF RECURSIVE MODEL THEORY Terrence S. MILLAR Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, U.S.A. Received 6 July 1976 A model is decidable if it has a decidable satisfaction predicate. To be more precise, let T be a decidable theory, let {0, I n < to} be an effective enumeration of all formula3 in L(T), and let 92 be a countable model of T. For any indexing E={a~ I i<to} of I~1, and any formula ~eL(T), let '~z' denote the result of substituting 'a{ for every free occurrence of 'x~' in q~, ~<o,. Then 92 is decidable just in case, for some indexing E of 192[, {n 192~0~ is a recursive set of integers. It is easy, to show that the decidability of a model does not depend on the choice of the effective enumeration of the formulas in L(T); we omit details. By a simple 'effectivizaton' of Henkin's proof of the completeness theorem [2] we have Fact 1. Every decidable theory has a decidable model. Assume next that T is a complete decidable theory and {On ln<to} is an effective enumeration of all formulas of L(T). A type F of T is recursive just in case {nlO, ~ F} is a recursive set of integers. Again, it is easy to see that the recursiveness of F does not depend .on which effective enumeration of L(T) is used. -
The Cyber Entscheidungsproblem: Or Why Cyber Can’T Be Secured and What Military Forces Should Do About It
THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier JCSP 41 PCEMI 41 Exercise Solo Flight Exercice Solo Flight Disclaimer Avertissement Opinions expressed remain those of the author and Les opinons exprimées n’engagent que leurs auteurs do not represent Department of National Defence or et ne reflètent aucunement des politiques du Canadian Forces policy. This paper may not be used Ministère de la Défense nationale ou des Forces without written permission. canadiennes. Ce papier ne peut être reproduit sans autorisation écrite. © Her Majesty the Queen in Right of Canada, as © Sa Majesté la Reine du Chef du Canada, représentée par represented by the Minister of National Defence, 2015. le ministre de la Défense nationale, 2015. CANADIAN FORCES COLLEGE – COLLÈGE DES FORCES CANADIENNES JCSP 41 – PCEMI 41 2014 – 2015 EXERCISE SOLO FLIGHT – EXERCICE SOLO FLIGHT THE CYBER ENTSCHEIDUNGSPROBLEM: OR WHY CYBER CAN’T BE SECURED AND WHAT MILITARY FORCES SHOULD DO ABOUT IT Maj F.J.A. Lauzier “This paper was written by a student “La présente étude a été rédigée par un attending the Canadian Forces College stagiaire du Collège des Forces in fulfilment of one of the requirements canadiennes pour satisfaire à l'une des of the Course of Studies. The paper is a exigences du cours. L'étude est un scholastic document, and thus contains document qui se rapporte au cours et facts and opinions, which the author contient donc des faits et des opinions alone considered appropriate and que seul l'auteur considère appropriés et correct for the subject. -
A Probabilistic Anytime Algorithm for the Halting Problem
Computability 7 (2018) 259–271 259 DOI 10.3233/COM-170073 IOS Press A probabilistic anytime algorithm for the halting problem Cristian S. Calude Department of Computer Science, University of Auckland, Auckland, New Zealand [email protected] www.cs.auckland.ac.nz/~cristian Monica Dumitrescu Faculty of Mathematics and Computer Science, Bucharest University, Romania [email protected] http://goo.gl/txsqpU The first author dedicates his contribution to this paper to the memory of his collaborator and friend S. Barry Cooper (1943–2015). Abstract. TheHaltingProblem,the most (in)famous undecidableproblem,has important applications in theoretical andapplied computer scienceand beyond, hencethe interest in its approximate solutions. Experimental results reportedonvarious models of computationsuggest that haltingprogramsare not uniformly distributed– running times play an important role.A reason is thataprogram whicheventually stopsbut does not halt “quickly”, stops ata timewhich is algorithmically compressible. In this paperweworkwith running times to defineaclass of computable probability distributions on theset of haltingprograms in ordertoconstructananytimealgorithmfor theHaltingproblem withaprobabilisticevaluationofthe errorofthe decision. Keywords: HaltingProblem,anytimealgorithm, running timedistribution 1. Introduction The Halting Problem asks to decide, from a description of an arbitrary program and an input, whether the computation of the program on that input will eventually stop or continue forever. In 1936 Alonzo Church, and independently Alan Turing, proved that (in Turing’s formulation) an algorithm to solve the Halting Problem for all possible program-input pairs does not exist; two equivalent models have been used to describe computa- tion by algorithms (an informal notion), the lambda calculus by Church and Turing machines by Turing. The Halting Problem is historically the first proved undecidable problem; it has many applications in mathematics, logic and theoretical as well as applied computer science, mathematics, physics, biology, etc. -
Classical First-Order Logic Software Formal Verification
Classical First-Order Logic Software Formal Verification Maria Jo~aoFrade Departmento de Inform´atica Universidade do Minho 2008/2009 Maria Jo~aoFrade (DI-UM) First-Order Logic (Classical) MFES 2008/09 1 / 31 Introduction First-order logic (FOL) is a richer language than propositional logic. Its lexicon contains not only the symbols ^, _, :, and ! (and parentheses) from propositional logic, but also the symbols 9 and 8 for \there exists" and \for all", along with various symbols to represent variables, constants, functions, and relations. There are two sorts of things involved in a first-order logic formula: terms, which denote the objects that we are talking about; formulas, which denote truth values. Examples: \Not all birds can fly." \Every child is younger than its mother." \Andy and Paul have the same maternal grandmother." Maria Jo~aoFrade (DI-UM) First-Order Logic (Classical) MFES 2008/09 2 / 31 Syntax Variables: x; y; z; : : : 2 X (represent arbitrary elements of an underlying set) Constants: a; b; c; : : : 2 C (represent specific elements of an underlying set) Functions: f; g; h; : : : 2 F (every function f as a fixed arity, ar(f)) Predicates: P; Q; R; : : : 2 P (every predicate P as a fixed arity, ar(P )) Fixed logical symbols: >, ?, ^, _, :, 8, 9 Fixed predicate symbol: = for \equals" (“first-order logic with equality") Maria Jo~aoFrade (DI-UM) First-Order Logic (Classical) MFES 2008/09 3 / 31 Syntax Terms The set T , of terms of FOL, is given by the abstract syntax T 3 t ::= x j c j f(t1; : : : ; tar(f)) Formulas The set L, of formulas of FOL, is given by the abstract syntax L 3 φ, ::= ? j > j :φ j φ ^ j φ _ j φ ! j t1 = t2 j 8x: φ j 9x: φ j P (t1; : : : ; tar(P )) :, 8, 9 bind most tightly; then _ and ^; then !, which is right-associative. -
Chapter 10: Symbolic Trails and Formal Proofs of Validity, Part 2
Essential Logic Ronald C. Pine CHAPTER 10: SYMBOLIC TRAILS AND FORMAL PROOFS OF VALIDITY, PART 2 Introduction In the previous chapter there were many frustrating signs that something was wrong with our formal proof method that relied on only nine elementary rules of validity. Very simple, intuitive valid arguments could not be shown to be valid. For instance, the following intuitively valid arguments cannot be shown to be valid using only the nine rules. Somalia and Iran are both foreign policy risks. Therefore, Iran is a foreign policy risk. S I / I Either Obama or McCain was President of the United States in 2009.1 McCain was not President in 2010. So, Obama was President of the United States in 2010. (O v C) ~(O C) ~C / O If the computer networking system works, then Johnson and Kaneshiro will both be connected to the home office. Therefore, if the networking system works, Johnson will be connected to the home office. N (J K) / N J Either the Start II treaty is ratified or this landmark treaty will not be worth the paper it is written on. Therefore, if the Start II treaty is not ratified, this landmark treaty will not be worth the paper it is written on. R v ~W / ~R ~W 1 This or statement is obviously exclusive, so note the translation. 427 If the light is on, then the light switch must be on. So, if the light switch in not on, then the light is not on. L S / ~S ~L Thus, the nine elementary rules of validity covered in the previous chapter must be only part of a complete system for constructing formal proofs of validity. -
Self-Referential Basis of Undecidable Dynamics: from the Liar Paradox and the Halting Problem to the Edge of Chaos
Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos Mikhail Prokopenko1, Michael Harre´1, Joseph Lizier1, Fabio Boschetti2, Pavlos Peppas3;4, Stuart Kauffman5 1Centre for Complex Systems, Faculty of Engineering and IT The University of Sydney, NSW 2006, Australia 2CSIRO Oceans and Atmosphere, Floreat, WA 6014, Australia 3Department of Business Administration, University of Patras, Patras 265 00, Greece 4University of Pennsylvania, Philadelphia, PA 19104, USA 5University of Pennsylvania, USA [email protected] Abstract In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical sys- tems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these com- putational frameworks. Some of these interconnections are well-known, while some are clarified in this study as a result of a fine-grained comparison between recursive formal systems, Turing machines, and Cellular Automata (CAs). In particular, we elaborate on the diagonalization argument applied to distributed computation carried out by CAs, illustrating the key elements of Godel’s¨ proof for CAs. The comparative analysis emphasizes three factors which underlie the capacity to generate undecidable dynamics within the examined computational frameworks: (i) the program-data duality; (ii) the potential to access an infinite computational medium; and (iii) the ability to im- plement negation. The considered adaptations of Godel’s¨ proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability. 1 Introduction It is well-known that there are deep connections between dynamical systems, algorithms, and formal systems. -
Model Theory
Class Notes for Mathematics 571 Spring 2010 Model Theory written by C. Ward Henson Mathematics Department University of Illinois 1409 West Green Street Urbana, Illinois 61801 email: [email protected] www: http://www.math.uiuc.edu/~henson/ c Copyright by C. Ward Henson 2010; all rights reserved. Introduction The purpose of Math 571 is to give a thorough introduction to the methods of model theory for first order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret. First order logic is the most important formal language and its model theory is a rich and interesting subject with significant applications to the main body of mathematics. Model theory began as a serious subject in the 1950s with the work of Abraham Robinson and Alfred Tarski, and since then it has been an active and successful area of research. Beyond the core techniques and results of model theory, Math 571 places a lot of emphasis on examples and applications, in order to show clearly the variety of ways in which model theory can be useful in mathematics. For example, we give a thorough treatment of the model theory of the field of real numbers (real closed fields) and show how this can be used to obtain the characterization of positive semi-definite rational functions that gives a solution to Hilbert’s 17th Problem. A highlight of Math 571 is a proof of Morley’s Theorem: if T is a complete theory in a countable language, and T is κ-categorical for some uncountable κ, then T is categorical for all uncountable κ. -
Halting Problems: a Historical Reading of a Paradigmatic Computer Science Problem
PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol Halting problems: a historical reading of a paradigmatic computer science problem Liesbeth De Mol1 and Edgar G. Daylight2 1 CNRS, UMR 8163 Savoirs, Textes, Langage, Universit´e de Lille, [email protected] 2 Siegen University, [email protected] 1 PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol and E.G. Daylight Introduction (1) \The improbable symbolism of Peano, Russel, and Whitehead, the analysis of proofs by flowcharts spearheaded by Gentzen, the definition of computability by Church and Turing, all inventions motivated by the purest of mathematics, mark the beginning of the computer revolution. Once more, we find a confirmation of the sentence Leonardo jotted despondently on one of those rambling sheets where he confided his innermost thoughts: `Theory is the captain, and application the soldier.' " (Metropolis, Howlett and Rota, 1980) Introduction 2 PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol and E.G. Daylight Introduction (2) Why is this `improbable' symbolism considered relevant in comput- ing? ) Different non-excluding answers... 1. (the socio-historical answers) studying social and institutional developments in computing to understand why logic, or, theory, was/is considered to be the captain (or not) e.g. need for logic framed in CS's struggle for disciplinary identity and independence (cf (Tedre 2015)) 2. (the philosophico-historical answers) studying history of computing on a more technical level to understand why and how logic (or theory) are in- troduced in the computing practices { question: is there something to com- puting which makes logic epistemologically relevant to it? ) significance of combining the different answers (and the respective approaches they result in) ) In this talk: focus on paradigmatic \problem" of (theoretical) computer science { the halting problem Introduction 3 PROGRAMme Autumn workshop 2018, Bertinoro L. -
Logic and Proof
Logic and Proof Computer Science Tripos Part IB Michaelmas Term Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2000 by Lawrence C. Paulson Contents 1 Introduction and Learning Guide 1 2 Propositional Logic 3 3 Proof Systems for Propositional Logic 12 4 Ordered Binary Decision Diagrams 19 5 First-order Logic 22 6 Formal Reasoning in First-Order Logic 29 7 Clause Methods for Propositional Logic 34 8 Skolem Functions and Herbrand’s Theorem 42 9 Unification 49 10 Applications of Unification 58 11 Modal Logics 65 12 Tableaux-Based Methods 70 i ii 1 1 Introduction and Learning Guide This course gives a brief introduction to logic, with including the resolution method of theorem-proving and its relation to the programming language Prolog. Formal logic is used for specifying and verifying computer systems and (some- times) for representing knowledge in Artificial Intelligence programs. The course should help you with Prolog for AI and its treatment of logic should be helpful for understanding other theoretical courses. Try to avoid getting bogged down in the details of how the various proof methods work, since you must also acquire an intuitive feel for logical reasoning. The most suitable course text is this book: Michael Huth and Mark Ryan, Logic in Computer Science: Modelling and Reasoning about Systems (CUP, 2000) It costs £18.36 from Amazon. It covers most aspects of this course with the ex- ception of resolution theorem proving. It includes material that may be useful in Specification and Verification II next year, namely symbolic model checking.