Download: Introduction To Computability by Frederick C. Hennie

1Incompleteness And Computability An Open Introduction To Godel's Theorems, Richard Zach

By Richard Zach, University of Calgary

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Incompleteness And Computability An Open Introduction To Godel's Theorems, Richard Zach” Metadata:

Title: ➤ Incompleteness And Computability An Open Introduction To Godel's Theorems, Richard Zach
Author: ➤ Richard Zach, University of Calgary
Language: English

Edition Identifiers:

Internet Archive ID: ➤ incompleteness-and-computability-an-open-introduction-to-godels-theorems-richard-zach

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The book is available for download in "texts" format, the size of the file-s is: 101.69 Mbs, the file-s for this book were downloaded 16 times, the file-s went public at Fri Jun 13 2025.

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2Enumerability, Decidability, Computability. An Introduction To The Theory Of Recursive Functions

By Hermes, Hans

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Enumerability, Decidability, Computability. An Introduction To The Theory Of Recursive Functions” Metadata:

Title: ➤ Enumerability, Decidability, Computability. An Introduction To The Theory Of Recursive Functions
Author: Hermes, Hans
Language: eng,ger

“Enumerability, Decidability, Computability. An Introduction To The Theory Of Recursive Functions” Subjects and Themes:

Subjects: Algorithms - Computable functions - Gödel's theorem - Recursive functions

Edition Identifiers:

Internet Archive ID: enumerabilitydec0000herm

Downloads Information:

The book is available for download in "texts" format, the size of the file-s is: 644.14 Mbs, the file-s for this book were downloaded 66 times, the file-s went public at Fri Jun 21 2019.

Available formats:
ACS Encrypted EPUB - ACS Encrypted PDF - Abbyy GZ - Cloth Cover Detection Log - DjVuTXT - Djvu XML - Dublin Core - EPUB - Item Tile - JPEG Thumb - JSON - LCP Encrypted EPUB - LCP Encrypted PDF - Log - MARC - MARC Binary - Metadata - OCR Page Index - OCR Search Text - PNG - Page Numbers JSON - Scandata - Single Page Original JP2 Tar - Single Page Processed JP2 ZIP - Text PDF - Title Page Detection Log - chOCR - hOCR -

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3Enumerability, Decidability, Computability; An Introduction To The Theory Of Recursive Functions

By Hermes, Hans

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Enumerability, Decidability, Computability; An Introduction To The Theory Of Recursive Functions” Metadata:

Title: ➤ Enumerability, Decidability, Computability; An Introduction To The Theory Of Recursive Functions
Author: Hermes, Hans
Language: eng,ger

“Enumerability, Decidability, Computability; An Introduction To The Theory Of Recursive Functions” Subjects and Themes:

Subjects: Algorithms - Computable functions - Gödel's theorem - Recursive functions

Edition Identifiers:

Internet Archive ID: enumerabilitydec0000herm_u9p7

Downloads Information:

The book is available for download in "texts" format, the size of the file-s is: 699.38 Mbs, the file-s for this book were downloaded 80 times, the file-s went public at Wed Jul 17 2019.

Available formats:
ACS Encrypted EPUB - ACS Encrypted PDF - Abbyy GZ - Cloth Cover Detection Log - DjVuTXT - Djvu XML - Dublin Core - EPUB - Item Tile - JPEG Thumb - JSON - LCP Encrypted EPUB - LCP Encrypted PDF - Log - MARC - MARC Binary - Metadata - OCR Page Index - OCR Search Text - PNG - Page Numbers JSON - Scandata - Single Page Original JP2 Tar - Single Page Processed JP2 ZIP - Text PDF - Title Page Detection Log - chOCR - hOCR -

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Find Enumerability, Decidability, Computability; An Introduction To The Theory Of Recursive Functions at online marketplaces:

Amazon: Audiable, Kindle and printed editions.

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4Automata, Computability, And Complexity- Introduction To Cryptography

By Scott Aaronson

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Automata, Computability, And Complexity- Introduction To Cryptography” Metadata:

Title: ➤ Automata, Computability, And Complexity- Introduction To Cryptography
Author: Scott Aaronson
Language: English

“Automata, Computability, And Complexity- Introduction To Cryptography” Subjects and Themes:

Subjects: Maths - Mathematics

Edition Identifiers:

Internet Archive ID: flooved1300

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The book is available for download in "texts" format, the size of the file-s is: 3.61 Mbs, the file-s for this book were downloaded 139 times, the file-s went public at Thu Nov 14 2013.

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5Automata, Computability, And Complexity- Introduction To Quantum

By Scott Aaronson

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Automata, Computability, And Complexity- Introduction To Quantum” Metadata:

Title: ➤ Automata, Computability, And Complexity- Introduction To Quantum
Author: Scott Aaronson
Language: English

“Automata, Computability, And Complexity- Introduction To Quantum” Subjects and Themes:

Subjects: Maths - Mathematics

Edition Identifiers:

Internet Archive ID: flooved1301

Downloads Information:

The book is available for download in "texts" format, the size of the file-s is: 2.41 Mbs, the file-s for this book were downloaded 113 times, the file-s went public at Thu Nov 14 2013.

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6Turing's World 3.0 For The Macintosh : An Introduction To Computability Theory

By Barwise, Jon and Etchemendy, John, 1952-

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Turing's World 3.0 For The Macintosh : An Introduction To Computability Theory” Metadata:

Title: ➤ Turing's World 3.0 For The Macintosh : An Introduction To Computability Theory
Authors: Barwise, JonEtchemendy, John, 1952-
Language: English

“Turing's World 3.0 For The Macintosh : An Introduction To Computability Theory” Subjects and Themes:

Subjects: ➤ Turing's world - Computer science - Logic, Symbolic and mathematical - Turing machines

Edition Identifiers:

Internet Archive ID: turingsworld30fo00barw

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The book is available for download in "texts" format, the size of the file-s is: 197.61 Mbs, the file-s for this book were downloaded 177 times, the file-s went public at Tue Jan 24 2012.

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7Computability, An Introduction To Recursive Function Theory

By Cutland, Nigel

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Computability, An Introduction To Recursive Function Theory” Metadata:

Title: ➤ Computability, An Introduction To Recursive Function Theory
Author: Cutland, Nigel
Language: English

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8The Language Of Machines : An Introduction To Computability And Formal Languages

By Floyd, Robert W

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“The Language Of Machines : An Introduction To Computability And Formal Languages” Metadata:

Title: ➤ The Language Of Machines : An Introduction To Computability And Formal Languages
Author: Floyd, Robert W
Language: English

“The Language Of Machines : An Introduction To Computability And Formal Languages” Subjects and Themes:

Subjects: Formal languages - Machine theory - Computable functions

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Internet Archive ID: languageofmachin0000floy

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9Introduction To Computability

By Hennie, Frederick C

About this Book x 1 Introduction to Incompleteness 1 1.1 Historical Background. . . . . . . . . . . . . . . 1 1.2 Definitions. . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview of Incompleteness Results. . . . . . . 13 1.4 Undecidability and Incompleteness. . . . . . . . 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 18 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Recursive Functions 20 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . 20 2.2 Primitive Recursion. . . . . . . . . . . . . . . . . 21 2.3 Composition. . . . . . . . . . . . . . . . . . . . . 24 2.4 Primitive Recursion Functions. . . . . . . . . . . 26 2.5 Primitive Recursion Notations. . . . . . . . . . . 30 2.6 Primitive Recursive Functions are Computable. . 31 2.7 Examples of Primitive Recursive Functions. . . . 32 2.8 Primitive Recursive Relations. . . . . . . . . . . 35 2.9 Bounded Minimization. . . . . . . . . . . . . . . 38 2.10 Primes. . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Sequences. . . . . . . . . . . . . . . . . . . . . . 41 2.12 Trees. . . . . . . . . . . . . . . . . . . . . . . . . 45 2.13 Other Recursions. . . . . . . . . . . . . . . . . . 46 2.14 Non-Primitive Recursive Functions. . . . . . . . 47 2.15 Partial Recursive Functions. . . . . . . . . . . . 49 2.16 The Normal Form Theorem. . . . . . . . . . . . 52 2.17 The Halting Problem. . . . . . . . . . . . . . . . 53 2.18 General Recursive Functions. . . . . . . . . . . . 55 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 55 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Arithmetization of Syntax 59 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . 59 3.2 Coding Symbols. . . . . . . . . . . . . . . . . . . 61 3.3 Coding Terms. . . . . . . . . . . . . . . . . . . . 63 3.4 Coding Formulas. . . . . . . . . . . . . . . . . . 65 3.5 Substitution. . . . . . . . . . . . . . . . . . . . . 67 3.6 Derivations in Natural Deduction. . . . . . . . . 68 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 74 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Representability in Q 76 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . 76 4.2 Functions Representable in Q are Computable. 79 4.3 The Beta Function Lemma. . . . . . . . . . . . . 81 4.4 Simulating Primitive Recursion. . . . . . . . . . 85 4.5 Basic Functions are Representable in Q . . . . . 86 4.6 Composition is Representable in Q . . . . . . . . 90 4.7 Regular Minimization is Representable in Q . . 92 4.8 Computable Functions are Representable in Q . 96 4.9 Representing Relations. . . . . . . . . . . . . . . 97 4.10 Undecidability. . . . . . . . . . . . . . . . . . . . 98 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 Incompleteness and Provability 102 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . 102 5.2 The Fixed-Point Lemma. . . . . . . . . . . . . . 104 5.3 The First Incompleteness Theorem. . . . . . . . 107 5.4 Rosser’s Theorem. . . . . . . . . . . . . . . . . . 109 5.5 Comparison with Gödel’s Original Paper. . . . . 111 5.6 The Derivability Conditions for PA . . . . . . . . 112 5.7 The Second Incompleteness Theorem. . . . . . 113 5.8 Löb’s Theorem. . . . . . . . . . . . . . . . . . . 116 5.9 The Undefinability of Truth. . . . . . . . . . . . 119 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 121 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 122 6 Models of Arithmetic 124 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . 124 6.2 Reducts and Expansions. . . . . . . . . . . . . . 125 6.3 Isomorphic Structures. . . . . . . . . . . . . . . 126 6.4 The Theory of a Structure. . . . . . . . . . . . . 129 6.5 Standard Models of Arithmetic. . . . . . . . . . 130 6.6 Non-Standard Models. . . . . . . . . . . . . . . 133 6.7 Models of Q . . . . . . . . . . . . . . . . . . . . . 134 6.8 Models of PA . . . . . . . . . . . . . . . . . . . . 137 6.9 Computable Models of Arithmetic. . . . . . . . 141 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 143 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Second-Order Logic 147 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . 147 7.2 Terms and Formulas. . . . . . . . . . . . . . . . 148 7.3 Satisfaction. . . . . . . . . . . . . . . . . . . . . 150 7.4 Semantic Notions. . . . . . . . . . . . . . . . . . 154 7.5 Expressive Power. . . . . . . . . . . . . . . . . . 154 7.6 Describing Infinite and Countable Domains. . . 156 7.7 Second-order Arithmetic. . . . . . . . . . . . . . 158 7.8 Second-order Logic is not Axiomatizable. . . . . 161 7.9 Second-order Logic is not Compact. . . . . . . . 162 7.10 The Löwenheim–Skolem Theorem Fails for Second-order Logic. . . . . . . . . . . . . . . . . 163 7.11 Comparing Sets. . . . . . . . . . . . . . . . . . . 163 7.12 Cardinalities of Sets. . . . . . . . . . . . . . . . . 165 7.13 The Power of the Continuum. . . . . . . . . . . 166

“Introduction To Computability” Metadata:

Title: Introduction To Computability
Author: Hennie, Frederick C
Language: English

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Subjects: Algorithms - Computational complexity - Recursive functions - Turing machines

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10Kurt Godel And Per Martin Lof A First Course In Logic An Introduction To Model Theory, Proof Theory, Computability, And Complexity ( Oxford Texts In Logic)

By Godel, Kurt

Acknowledgments - Florida Southern College provided a most pleasant and hospitable setting for the writing of this book. Thanks to all of my friends and colleagues at the college. In particular, I thank colleague David Rose and student Biljana Cokovic for reading portions of the manuscript and offering helpful feedback. I thank my colleague Mike Way for much needed technological assistance. This book began as lecture notes for a course I taught at the University of Maryland.

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Title: ➤ Kurt Godel And Per Martin Lof A First Course In Logic An Introduction To Model Theory, Proof Theory, Computability, And Complexity ( Oxford Texts In Logic)
Author: Godel, Kurt
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“Kurt Godel And Per Martin Lof A First Course In Logic An Introduction To Model Theory, Proof Theory, Computability, And Complexity ( Oxford Texts In Logic)” Subjects and Themes:

Subjects: Logique - Mathematique - Philosophie

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