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

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