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Recursion theory for metamathematics

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"This work is a sequel to the author's Godel's Incompleteness Theorems, though it can be read independently by anyone familiar with Godel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new  Read more...

Abstract:
In 1931, Princeton mathematician Kurt Godel startled the scientific world with his 'Theorem of Undecidability', which showed that some statements in mathematics are inherently 'undecidable'. This  Read more...

Author(s): Smullyan, Raymond M
Series: Oxford logic guides 22
Publisher: Oxford University Press
Year: 1993

Language: English
Pages: 163
City: New York
Tags: Recursion theory.;Recursivité, Théorie de la.;Recursie.;Récursivité, théorie de la.;Mathematical logic

Content: 0 Prerequisites (starting p. 1) --
I Some General Incompleteness Theorems (starting p. 1) --
II Arithmetic, [Sigma[subscript 0]] and R.E. Relations (starting p. 8) --
III Shepherdson's Theorems (starting p. 19) --
I Recursive Enumerability and Recursivity (starting p. 23) --
I Some Basic Closure Properties (starting p. 23) --
II Recursive Pairing Functions (starting p. 29) --
III Representability and Recursive Enumerability (starting p. 32) --
II Undecidability and Recursive Inseparability (starting p. 35) --
I Undecidability (starting p. 35) --
II Recursive Inseparability (starting p. 40) --
III Indexing (starting p. 45) --
I The Enumeration Theorem (starting p. 45) --
II The Iteration Theorem (starting p. 47) --
III Effective Separation (starting p. 52) --
IV Generative Sets and Creative Systems (starting p. 54) --
V Double Generativity and Complete Effective Inseparability (starting p. 63) --
I Complete Effective Inseparability (starting p. 63) --
II Double Universality (starting p. 71) --
III Double Generativity (starting p. 72) --
VI Universal and Doubly Universal Systems (starting p. 77) --
I Universality (starting p. 78) --
II Double Universality (starting p. 80) --
VII Shepherdson Revisited (starting p. 84) --
VIII Recursion Theorems (starting p. 89) --
I Weak Recursion Theorems (starting p. 90) --
II The Strong Recursion Theorem (starting p. 93) --
III An Extended Recursion Theorem (starting p. 97) --
IX Symmetric and Double Recursion Theorems (starting p. 99) --
I Double Recursion Theorems (starting p. 99) --
II A Symmetric Recursion Theorem (starting p. 103) --
III Double Recursion With a Pairing Function (starting p. 104) --
IV Further Topics (starting p. 107) --
X Productivity and Double Productivity (starting p. 113) --
I Productivity and Double Productivity (starting p. 113) --
XI Three Special Topics (starting p. 121) --
I Uniform Reducibility (starting p. 121) --
II Pseudo-Uniform Reducibility (starting p. 126) --
III Some Feeble Partial Functions (starting p. 129) --
XII Uniform Godelization (starting p. 134) --
I The Sentential Recursion Property (starting p. 134) --
II DSR and Semi-DSR Systems (starting p. 136) --
III Rosser Fixed-Point Properties and Uniform Incompletability (starting p. 141) --
IV Finale (starting p. 147) --
References (starting p. 151) --
Index (starting p. 153)