Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory Robert I. Soare∗ January 3, 2009 Contents 1 Introduction 4 1.1 Terminology: Incompleteness and Incomputability . 4 1.2 The Goal of Incomputability not Computability . 5 1.3 Computing Relative to an Oracle or Database . 5 1.4 Continuous Functions and Calculus . 6 2 Origins of Computability and Incomputability 7 2.1 G¨odel'sIncompleteness Theorem . 7 2.2 Alonzo Church . 8 2.3 Herbrand-G¨odelRecursive Functions . 9 2.4 Stalemate at Princeton Over Church's Thesis . 10 2.5 G¨odel'sThoughts on Church's Thesis . 11 3 Turing Breaks the Stalemate 11 3.1 Turing Machines and Turing's Thesis . 11 3.2 G¨odel'sOpinion of Turing's Work . 13 3.2.1 G¨odel[193?] Notes in Nachlass [1935] . 14 3.2.2 Princeton Bicentennial [1946] . 15 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18{23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene, Alan Turing, Kurt G¨odel, Emil Post, computability, incomputability, undecidability, Church-Turing Thesis, Post-Turing Thesis on relative computability, com- putable approximations, Limit Lemma, effectively continuous functions, computability in analysis, strong reducibilities. Thanks are due to C.G. Jockusch, Jr., P. Cholak, and T. Slaman for corrections and suggestions. 1 3.2.3 The Flaw in Church's Thesis . 16 3.2.4 G¨odelon Church's Thesis . 17 3.2.5 G¨odel'sLetter to Kreisel [1968] .
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