DOCSLIB.ORG

Kurt Gödel and Computability Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Kurt Gödel and Computability Theory Kurt Gödel and Computability Theory Richard Zach University of Calgary, Canada www.ucalgary.ca/∼rzach/ CiE 2006 July 5, 2006 Richard Zach Kurt Gödel and Computability Theory Importance of Logical Pioneers to CiE Wilhelm Ackermann Paul Bernays Alonzo Church Gerhard Gentzen Kurt Gödel Stephen Kleene Andrei Kolmogorov Rosza Péter Emil Post J. Barkley Rosser Kurt Schütte Thoralf Skolem Alfred Tarski Alan Turing John von Neumann Richard Zach Kurt Gödel and Computability Theory Importance of Logical Pioneers to CiE Wilhelm Ackermann 0 Paul Bernays 0 Alonzo Church 1 Gerhard Gentzen 2 Kurt Gödel 3 Stephen Kleene 1 Andrei Kolmogorov 0 Rosza Péter 1 Emil Post 1 J. Barkley Rosser 0 Kurt Schütte 1 Thoralf Skolem 1 Alfred Tarski 0 Alan Turing 8 John von Neumann 0 Richard Zach Kurt Gödel and Computability Theory Gödel’s Legacy for Computability Completeness of the predicate calculus. Incompleteness of systems including arithmetic. Work on the decision problem (decidability of Gödel-Kalmár-Schütte class ∃∗∀∀∃∗). Herbrand-Gödel definition of general recursive functions. Functions reckonable in a formal system. Gödel-Gentzen translation of classical to intuitionistic logic/arithmetic. P.r. functionals of finite type (Dialectica interpretation). Richard Zach Kurt Gödel and Computability Theory Arithmetization of Syntax Gödel numbering of syntax (formulas, proofs) Showed that important functions and relations on syntax (“is a formula”, “is a proof”, substitution) are primitive recursive Showed that primitive recursive functions are numeralwise representable in Principia Showed that formulas which numeralwise represent p.r. functions are arithmetical (Gödel’s β-function, arithmetical coding of sequences) Richard Zach Kurt Gödel and Computability Theory Arithmetization of Syntax Gödel numbering of syntax (formulas, proofs) Showed that important functions and relations on syntax (“is a formula”, “is a proof”, substitution) are primitive recursive Showed that primitive recursive functions are numeralwise representable in Principia Showed that formulas which numeralwise represent p.r. functions are arithmetical (Gödel’s β-function, arithmetical coding of sequences) Richard Zach Kurt Gödel and Computability Theory Arithmetization of Syntax Gödel numbering of syntax (formulas, proofs) Showed that important functions and relations on syntax (“is a formula”, “is a proof”, substitution) are primitive recursive Showed that primitive recursive functions are numeralwise representable in Principia Showed that formulas which numeralwise represent p.r. functions are arithmetical (Gödel’s β-function, arithmetical coding of sequences) Richard Zach Kurt Gödel and Computability Theory Arithmetization of Syntax Gödel numbering of syntax (formulas, proofs) Showed that important functions and relations on syntax (“is a formula”, “is a proof”, substitution) are primitive recursive Showed that primitive recursive functions are numeralwise representable in Principia Showed that formulas which numeralwise represent p.r. functions are arithmetical (Gödel’s β-function, arithmetical coding of sequences) Richard Zach Kurt Gödel and Computability Theory Church’s “Foundation of Logic” Church, “A set of postulates for the foundation of logic”, Annals of Mathematics 1932, 1933. Developed 1929–1931 Gave course on it at Princeton in Fall 1931, Kleene took notes John von Neumann gave talk on Gödel’s 1931 incompleteness results Question: did Gödel’s results apply to Church’s system? Kleene tasked with developing Peano arithmetic in Church’s system Richard Zach Kurt Gödel and Computability Theory Consistency of Church’s System [. I]t remains barely possible that a proof of freedom from contradiction for my system can be found somewhat along the lines suggested by Hilbert. I have, in fact, made several unsuccessful attempts to do this. Dr. von Neumann called my attention last Fall to your paper entitled “Über formal unentscheidbare Sätze der Principia Mathematica.” I have been unable to see, however, that your conclusions in §4 of this paper [on the second incompleteness theorem] apply to my system. Possibly your argument can be modified so as to make it apply to my system, but I have not been able to find such a modification of your argument. (Church to Gödel, July 27, 1932.) Richard Zach Kurt Gödel and Computability Theory Peano Arithmetic in Church’s System Kleene, “A theory of positive integers in formal logic”, American J. Mathematics 1935 (work done in 1932) Definitions of numerals, +, ×, etc. as λ-terms Every primitive recursive function is λ-definable Used arithmetization of syntax to show that question of derivability in a formal system (e.g., Principia) is equivalent to question of whether a certain λ-term has a normal form Richard Zach Kurt Gödel and Computability Theory Gödel’s 1934 Princeton Course February–May 1934 Church, Rosser, Kleene attended Definition of general recursive functions Richard Zach Kurt Gödel and Computability Theory Inconsistency of Church’s System Kleene and Rosser, “The inconsistency of certain formal logics,” Annals of Mathematics 1935 (submitted 1934) Uses Gödel’s arithmetization of syntax to derive contradiction. Richard Zach Kurt Gödel and Computability Theory Church’s Theorem Church, “An unsolvable problem of elementary number theory,” American J. Mathematics 1936 λ-terms with normal form not λ-definable. Together with Gödel’s arithmetization of syntax allows representation in the system of “term t has a normal form” Church and Kleene’s result that λ-definability equivalent to general recursiveness (1935), Church’s Thesis (effectively computable = general recursive), yields: unsolvability of decision problem of Principia. Reduced to decision problem of predicate calculus in “A note on the Entscheidungsproblem”, JSL 1936. Richard Zach Kurt Gödel and Computability Theory Recursive Function Theory Kleene, “General recursive functions of natural numbers”, Mathematische Annalen 1936 Systematic study of general recursive functions Arithmetization à la Gödel of computations Kleene’s T -predicate, indexes for recursive functions Normal form theorem, µ-recursion Construction of non-recursive functions Recursion theorem, s-m-n theorem 1938 Richard Zach Kurt Gödel and Computability Theory Influence of Gödel on Recursion Theory Prompted Church and Kleene to develop arithmetic in Church’s system Prompted development of λ-definability Methods used essentially to show Church’s system inconsistent λ-definability of p.r. functions prompted initial tentative formulation (1933) of Church’s Thesis Equivalence of λ-definability and general recursiveness prompted Church’s statement of Thesis in print (1936) Arithmetization of provability (via λ-definability) central step in Church’s Theorem Arithmetization central for basics of recursive function theory (T -predicate) Richard Zach Kurt Gödel and Computability Theory Kleene on Gödel After the colloquium [by von Neumann in the fall of 1931], Church’s course continued uninterruptedly concentrating on his formal system; but on the side we all read Gödel’s paper, which to me opened up a whole new world of fascinating ideas and perspectives. Richard Zach Kurt Gödel and Computability Theory.
Load more

Recommended publications
  • Arxiv:1504.04798V1 [Math.LO] 19 Apr 2015 of Principia Squarely in an Empiricist Framework
    HEINRICH BEHMANN'S 1921 LECTURE ON THE DECISION PROBLEM AND THE ALGEBRA OF LOGIC PAOLO MANCOSU AND RICHARD ZACH Abstract. Heinrich Behmann (1891{1970) obtained his Habilitation under David Hilbert in G¨ottingenin 1921 with a thesis on the decision problem. In his thesis, he solved|independently of L¨owenheim and Skolem's earlier work|the decision prob- lem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in G¨ottingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included. x1. Behmann's Career. Heinrich Behmann was born January 10, 1891, in Bremen. In 1909 he enrolled at the University of T¨ubingen. There he studied mathematics and physics for two semesters and then moved to Leipzig, where he continued his studies for three semesters. In 1911 he moved to G¨ottingen,at that time the most important center of mathematical activity in Germany. He volunteered for military duty in World War I, was severely wounded in 1915, and returned to G¨ottingenin 1916. In 1918, he obtained his doctorate with a thesis titled The Antin- omy of Transfinite Numbers and its Resolution by the Theory of Russell and Whitehead [Die Antinomie der transfiniten Zahl und ihre Aufl¨osung durch die Theorie von Russell und Whitehead] under the supervision of David Hilbert [Behmann 1918].
    [Show full text]
  • Henkin's Method and the Completeness Theorem
    Henkin's Method and the Completeness Theorem Guram Bezhanishvili∗ 1 Introduction Let L be a first-order logic. For a sentence ' of L, we will use the standard notation \` '" for ' is provable in L (that is, ' is derivable from the axioms of L by the use of the inference rules of L); and \j= '" for ' is valid (that is, ' is satisfied in every interpretation of L). The soundness theorem for L states that if ` ', then j= '; and the completeness theorem for L states that if j= ', then ` '. Put together, the soundness and completeness theorems yield the correctness theorem for L: a sentence is derivable in L iff it is valid. Thus, they establish a crucial feature of L; namely, that syntax and semantics of L go hand-in-hand: every theorem of L is a logical law (can not be refuted in any interpretation of L), and every logical law can actually be derived in L. In fact, a stronger version of this result is also true. For each first-order theory T and a sentence ' (in the language of T ), we have that T ` ' iff T j= '. Thus, each first-order theory T (pick your favorite one!) is sound and complete in the sense that everything that we can derive from T is true in all models of T , and everything that is true in all models of T is in fact derivable from T . This is a very strong result indeed. One possible reading of it is that the first-order formalization of a given mathematical theory is adequate in the sense that every true statement about T that can be formalized in the first-order language of T is derivable from the axioms of T .
    [Show full text]
  • • Hermann Weyll, 1927. Comments on Hilbert's
    Hermann Weyll, 1927. Comments on Hilbert's second lecture on the • foundations of mathematics. Response to Hilbert's 1927 lecture. Weyll defends Poincar´ewho had a problem (circularity in the justification) with the with the metamathematical use of mathematical induction. Most important point: in constructive mathematics, the rule of generalization and of induction blend. Paul Bernays, 1927. Appendix to Hilbert's lecture \The foundations • of mathematics". This paper by Bernays presents a proof of Ackermann (second version) of the consistency of a certain system. The proof idea comes from Hilbert but is here formally worked out in a more general setting. The setting is not so general that it comprises all of analysis. Luitzen Egbertus Jan Brouwer, 1927a. Intuitionistic reflections on • formalism. This text is the first paragraph of a longer paper by Brouwer. He comes back to an earlier statement of his that dialogue between logicism (?) and intuitionism is excluded. In this paper his list four point on which formalism and intuitionism could enter a dialogue. The relevant (subjective!) points concern the relation between finite metamathematics and some parts of intuitionism. Wilhelm Ackermann, 1928. On HIlbert's constructoin of the real num- • bers/ I cannot really match the title with the content of the introduction. Probably Hilbert uses primitive recursive functions in constructing his real numbers. Ackermann presents a study of a generalized version of the recursion schema by allowing functions of lower type to occur in the re- cursing definitions. A function is introduced which is shown to be larger than every primitive recursive function. This function can be represented by a type-1 recursion simultaious on two variables.
    [Show full text]
  • Kurt Gödel Metamathematical Results on Formally Undecidable Propositions: Completeness Vs
    Kurt Gödel Metamathematical results on formally undecidable propositions: Completeness vs. Incompleteness Motto: The delight in seeing and comprehending is the most beautiful gift of nature. (A.Einstein) 1. Life and work1 Kurt Gödel was a solitary genius, whose work influenced all the subsequent developments in mathematics and logic. The striking fundamental results in the decade 1929-1939 that made Gödel famous are the completeness of the first-order predicate logic proof calculus, the incompleteness of axiomatic theories containing arithmetic, and the consistency of the axiom of choice and the continuum hypothesis with the other axioms of set theory. During the same decade Gödel made other contributions to logic, including work on intuitionism and computability, and later, under the influence of his friendship with Einstein, made a fundamental contribution to the theory of space-time. In this article I am going to summarize the most outstanding results on incompleteness and undecidability that changed the fundamental views of modern mathematics. Kurt Friedrich Gödel was born 28 April 1906, the second son of Rudolf and Marianne (Handschuh) Gödel, in Brno, Pekařská 5,2 in Moravia, at that time the Austrio-Hungarian province, now a part of the Czech Republic. This region had a mixed population that was predominantly Czech with a substantial German-speaking minority, to which Gödel’s parents belonged. Following the religion of his mother rather than his “old-catholic” father, the Gödels had Kurt baptized in the Lutheran church. In 1912, at the age of six, he was enrolled in the Evangelische Volksschule, a Lutheran school in Brno. Gödel’s ethnic patrimony is neither Czech nor Jewish, as is sometimes believed.
    [Show full text]
  • 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
    [Show full text]
  • Hilbert's Program Then And
    Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory).
    [Show full text]
  • Ackermann Function 1 Ackermann Function
    Ackermann function 1 Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication[1] of his function (which had three nonnegative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function, is defined as follows for nonnegative integers m and n: Its value grows rapidly, even for small inputs. For example A(4,2) is an integer of 19,729 decimal digits.[2] History In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation. Both Sudan and Ackermann are credited[3] with discovering total computable functions (termed simply "recursive" in some references) that are not primitive recursive. Sudan published the lesser-known Sudan function, then shortly afterwards and independently, in 1928, Ackermann published his function . Ackermann's three-argument function, , is defined such that for p = 0, 1, 2, it reproduces the basic operations of addition, multiplication, and exponentiation as and for p > 2 it extends these basic operations in a way
    [Show full text]
  • Henkin's Method and the Completeness Theorem
    Henkin's Method and the Completeness Theorem Guram Bezhanishvili∗ 1 Introduction Let A be a first-order alphabet and L be the first-order logic in the alphabet A. For a sentence ' in the alphabet A, we will use the standard notation \` '" for ' is provable in L (that is, ' is derivable from the axioms of L by the use of the inference rules of L); and \j= '" for ' is valid (that is, ' is satisfied in every interpretation of L). The soundness theorem for L states that if ` ', then j= '; and the completeness theorem for L states that if j= ', then ` '. Put together, the soundness and completeness theorems yield the correctness theorem for L: a sentence is derivable in L iff it is valid. Thus, they establish a crucial feature of L; namely, that syntax and semantics of L go hand-in-hand: every theorem of L is a logical law (can not be refuted in any interpretation of L), and every logical law can actually be derived in L. In fact, a stronger version of this result is also true. For each first-order theory T and a sentence ' (in the language of T ), we have that T ` ' iff T j= '. Thus, each first-order theory T (pick your favorite one!) is sound and complete in the sense that everything that we can derive from T is true in all models of T , and everything that is true in all models of T is in fact derivable from T . This is a very strong result indeed. One possible reading of it is that the first-order formalization of a given mathematical theory is adequate in the sense that every true statement about the theory can be derived from the axioms of T .
    [Show full text]
  • On Godel's Way In: the Influence of Rudolf Carnap
    On Godel's Way In: The Influence of Rudolf Carnap The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Goldfarb, Warren. 2005. On Godel's way in: The influence of Rudolf Carnap. Bulletin of Symbolic Logic 11, no. 2: 185-193. Published Version http://dx.doi.org/10.2178/bsl/1120231629 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3153305 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA On Godel's Way in: The Influence of Rudolf Carnap Author(s): Warren Goldfarb Source: The Bulletin of Symbolic Logic, Vol. 11, No. 2 (Jun., 2005), pp. 185-193 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/1556748 Accessed: 24/06/2009 18:20 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl.
    [Show full text]
  • The Decision Problem and the Development of Metalogic
    Historical Background Behmann’s Solution for the Monadic Class Decision Problems after Behmann Historical Discussion The Decision Problem and the Development of Metalogic Richard Zach University of Calgary, Canada Department of Philosophy and Calgary Institute for the Humanities www.ucalgary.ca/∼rzach/ ASL Annual Meeting March 12, 2007 1/38 Richard Zach Decision Problem and Development of Metalogic Historical Background Behmann’s Solution for the Monadic Class Decision Problems after Behmann Historical Discussion Outline 1 Historical Background 2 Behmann’s Solution for the Monadic Class 3 Decision Problems after Behmann 4 Historical Discussion 2/38 Richard Zach Decision Problem and Development of Metalogic Historical Background Introduction Behmann’s Solution for the Monadic Class Dramatis personae Decision Problems after Behmann Early Work on Decidability Historical Discussion The Decision Problem Introduction Decision Problem: find a procedure which decides after finitely many steps if a given formula is valid or not One of the fundamental problems for the development of modern logic in the 1920s and 1930s Intimately tied up with Hilbert’s philosophical project development of model theory development of notion of decision procedure/computation 3/38 Richard Zach Decision Problem and Development of Metalogic Historical Background Introduction Behmann’s Solution for the Monadic Class Dramatis personae Decision Problems after Behmann Early Work on Decidability Historical Discussion The Decision Problem David Hilbert, 1862–1943 Studied mathematics
    [Show full text]
  • Inference Rules (Pt
    Inference Rules (Pt. II) Important Concepts A subderivation is a procedure through which we make a new assumption to derive a sentence of TL we couldn’t have otherwise derived. All subderivations are initiated with an assumption for the application of a Subderivations subderivational rule of inference, i.e., ⊃I, ~I, ~E, ∨E, ≡I. All subderivation rules, however, must eventually close the assumption with which it was initated. Rules of Inference Conditional Introduction (⊃I) P Where P and Q are Q meta-variables ranging over declarative P ⊃ Q sentences... Negation Introduction (~I) P Where P and Q are Q meta-variables ranging over declarative ~Q sentences... ~P Negation Elimination (~E) ~P Where P and Q are Q meta-variables ranging over declarative ~Q sentences... P P ∨ Q Disjunction Elimination P (∨E) R Where P and Q are . meta-variables ranging Q over declarative R sentences... R Biconditional P Introduction (≡I) Q . Where P and Q are Q meta-variables ranging over declarative P sentences... P ≡ Q A derivation in SD is a series of sentences of TL, each of which is either an assumption or is obtained from Derivations in SD previous sentences by one of the rules of SD. Fitch Notation is the notational system we will use for constructing formal proofs. Fitch-style proofs arrange the sequence Fitch Notation of sentences that make up the proof into rows and use varying degrees of indentation for the assumptions made throughout the proof. Premise(s) Assumption ... ... Inference(s) Justification ... ... Conclusion Justification Storytime! David Hilbert publishes a list of 23 unsolved problems in Mathematics, 1900 Among the problems was the continuing puzzle over Logicism..
    [Show full text]
  • Presburger's Article on Integer Arithmetic
    @TechReport{Stansifer:1884:PAIA, author="Ryan Stansifer", title= "Presburger’s Article on Integer Airthmetic: Remarks and Translation", number="TR84-639", institution="Cornell University, Computer Science Department", url="http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR84-639", month= "September", year= "1984", } Presburger’s Article on Integer Arithmetic: Remarks and Translation Ryan Stansifer An early chapter in the development of decision procedures concerns the theory of Presburger arithmetic. The original article presenting the theory was published in German in 1930 under the title “Uber¨ die Vollst¨andigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt.” My translation of this article appears here. The translation is preceded by remarks about the historical circumstances surrounding the paper and about the paper itself. The article was written by Moj˙zesz Presburger, a Polish student of mathematics, and presented at a conference in Warsaw in 1929. In it Presburger showed that the part of number theory which uses only the addition function is complete, that is, every formula or its negation is true. This fragment of number theory has come to be known as Presburger arithmetic. Although the motivation of his paper was to prove the theory complete, the method of proof is constructive and yields a decision procedure or algorithm with which every formula of Presburger arithmetic can be determined to be true or false. Probably the earliest theorem-proving program ever written for the computer employed Presburger’s algorithm to prove theorems of Presburger arithmetic. Martin Davis wrote the program in the summer of 1954 for an electronic digital computer with a memory of only 1024 words (Davis 1960).
    [Show full text]