Kurt Gödel and Computability Theory

Transcription

1 University of Calgary, Canada rzach/ CiE 2006 July 5, 2006

2 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

3 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

4 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).

5 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)

6 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)

7 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)

8 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)

9 Church s Foundation of Logic Church, A set of postulates for the foundation of logic, Annals of Mathematics 1932, Developed 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

10 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.)

11 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

12 Gödel s 1934 Princeton Course February May 1934 Church, Rosser, Kleene attended Definition of general recursive functions

13 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.

14 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.

15 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

16 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)

17 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.