Recursion Theory for Metamathematics [NOOK Book]

Overview

This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel'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 results in the field.

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Recursion Theory for Metamathematics

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Overview

This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel'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 results in the field.

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Editorial Reviews

From the Publisher
"A self-contained exposition, it presumes neither a reading of the previous volume . . . nor a prior knowledge of recursion theory. The reader with some familiarity with both recursion theory and logic should find this an excellent source for the interconnections between them." —Choice

"Smullyan is not only an outstanding authority on this subject, but is also a skilled pedagogue, with a special talent for formulating simple riddles, which illuminate this very difficult and profound subject. Smullyan has made an important contribution toward the wider understanding of the work of Godel and his followers. . . . Smullyan plays a significant role in the further development of mathematical logic and the elucidation of its relation to metamathematics. He continues to be one of the foremost popularizers of the subject." —American Scientist

". . . an interesting presentation of recursion theory from the point of view of its applications in metamathematics, indicating many interrelations between various notions and properties. It will certainly be studied carefully and referred to by students and specialists alike." —Roman Murawski, Mathematical Reviews

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Product Details

  • ISBN-13: 9780195344813
  • Publisher: Oxford University Press
  • Publication date: 1/28/1993
  • Series: Oxford Logic Guides
  • Sold by: Barnes & Noble
  • Format: eBook
  • File size: 2 MB

Table of Contents

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