Simple type theory: a practical logic for expressing and reasoning about mathematical ideas
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham
Birkhäuser
[2023]
Cham Springer International Publishing |
| Schriftenreihe: | Computer science foundations and applied logic
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 295 Seiten Diagramme |
| ISBN: | 9783031211140 9783031211119 |
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| 245 | 1 | 0 | |a Simple type theory |b a practical logic for expressing and reasoning about mathematical ideas |c by William M. Farmer |
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| 650 | 4 | |a Computational Complexity | |
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| 650 | 4 | |a Set Theory | |
| 650 | 4 | |a Computer science | |
| 650 | 4 | |a Mathematical logic | |
| 650 | 4 | |a Computational complexity | |
| 650 | 4 | |a Reasoning | |
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Datensatz im Suchindex
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Contents Preface 1 Introduction Summary of the Contents. 2 Answers to Readers’ Questions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3 Why Logic?. Why a Practical Logic?. . . Why Simple Type Theory?. Why not First-Order Logic?. Why not Set Theory?. Why not Dependent Type Theory?. Why Undefinedness?. Why Model Theory instead of Proof Theory?. Preliminary Concepts What is Mathematics?. Mathematical Values. 3.2.1 Sets. 3.2.2 Sequences. 3.2.3 Relations. 3.2.4 Functions. 3.2.5 Boolean Values and Predicates. 3.3 Binders. 3.4
Undefinedness. 3.5 Mathematical Structures. 3.6 Examples of Mathematical Structures. 3.7 Conclusions. 3.8 Exercises . 3.1 3.2 xiii 1 3 6 6 6 7 8 9 10 11 11 13 13 15 16 17 18 19 21 22 25 27 30 32 32 vii
CONTENTS viii 4 Syntax 5 4.1 Notation. 4.2 Symbols. 4.3 Types. 4.4 Expressions. 4.5 Bound and Free Variables. 4.6 Substitution. 4.7 Languages. 4.8 Conclusions. 4.9 Exercises . 42 42 44 44 45 Semantics 47 Interpretations. General Models. Finite General Models. Standard Models. 47 48 52 53 5.5 Satisfiability, Validity, and Semantic Consequence. 54 5.6 5.7 5.8 5.9 5.10 5.11 Isomorphic General Models. Expansion of a General
Model. Standard vs. General Semantics. Examples of Standard Models. Conclusions. Exercises . 56 57 58 59 62 62 5.1 5.2 5.3 5.4 6 Additional Notation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Boolean Operators. Binary Operators. Quantifiers . Definedness. Sets. Tuples. Functions. Miscellaneous Notation. Quasitypes and Dependent Quasitypes . Conclusions. Exercises . 7 Beta-Reduction and Substitution
7.1 Beta-Reduction. 7.2 Universal Instantiation. 64 64 67 68 69 70 70 70 70 72 75 75 77 77 g0
CONTENTS 7.3 7.4 7.5 7.6 Invalid Beta-Reduction. Alpha-Conversion. Conclusions. Exercises . ix 81 82 83 83 8 Proof Systems 84 8.1 Background. 84 8.2 A Proof System for Alonzo. 86 8.2.1 Axioms. 86 8.2.2 Rules of Inference . 88 8.2.3 Proofs. 88 8.3 Soundness. 89 8.4 Rugai General Models. 89 8.5 Completeness. 90 8.6 Conclusions. 93 8.7 Exercises . 93 9 Theories 94 9.1 Axiomatic Theories. 94 9.2 Theory
Extensions. 101 9.3 Conservative Theory Extensions . 104 9.4 Categorical Theories. Ш 9.5 Complete Theories. H$ 9.6 Fundamental Form of a Mathematical Problem. 120 9.7 Model Theory. 121 9.8 Conclusions. 122 9.9 Exercises . 12Ί 10 Sequences 12^ 10.1 Systems of Natural Numbers.12? 10.2 Notation for Sequences. 12$ 10.3 Conclusions . . 1$θ 10.4 Exercises . 1$θ 11 Developments ^1 11.1 Theory Developments. 1^1 11.2 Development of Natural Number Arithmetic. ■ · · 134 11.2.1 Basic Definitions and Theorems.I34 11.2.2 Commutative Semiring. 11.2.3 Weak Total
Order.136
CONTENTS x 11.2.4 Divides Lattice. ^^θ 11.3 Conclusions. 138 11.4 Exercises .138 12 Real Number Mathematics 139 12.1 Complete Ordered Fields . 140 12.2 Alternatives to the Construction of COF. 143 12.3 Development of Real Number Mathematics. 145 12.3.1 Some Basic Definitions and Theorems. 145 12.3.2 Naturals, Integers, and Rationals. 146 12.3.3 Iterated Sum and Product Operators. 147 12.3.4 Calculus. 148 12.3.5 Euclidean Space. 150 12.4 Skolem’s Paradox. 153 12.5 Conclusions. 154 12.6 Exercises . 154 13 Morphisms 156 13.1 A Motivating Example. 156 13.2 The Little Theories
Method. 158 13.3 Theory Morphisms. 159 13.3.1 Theory Translations. 159 13.3.2 Morphism Theorem . 164 13.3.3 Examples of TheoryMorphisms. 167 13.3.4 Faithful Theory Morphisms. 175 13.4 Development Morphisms. 177 13.4.1 Development Translations. 178 13.4.2 Transportations. 182 13.5 Mathematics Libraries. 187 13.5.1 Theory Graphs. 187 13.5.2 Development Graphs. 188 13.5.3 Realm Graphs . 188 13.6 Theory Graph Combinatore. 189 13.7 Conclusions. . 13.8 Exercises . 14 Alonzo Variants 14.1 Alonzo with Indefinite Description. 14.1.1
Introduction . 14.1.2 Syntax. 196 1gß լցք 1Q7
CONTENTS xi 14.1.3 Semantics. 197 14.1.4 Proof System. 197 14.1.5 Theorems. 198 14.2 Alonzo with Sorts . 199 14.2.1 Introduction . 199 14.2.2 Syntax. 199 14.2.3 Semantics. 201 14.2.4 Proof System. 203 14.2.5 Theorems . . 204 14.3 Alonzo with Quotation and Evaluation. 204 14.3.1 Introduction . 204 14.3.2 Syntax. 205 14.3.3 Semantics. 207 14.3.4 Proof System. 208 14.3.5 Theorems. 208 14.4 Conclusions. 209 15 Software Support 210 15.1 Basis
Support. 211 15.2 Advanced Support. 212 15.2.1 Organization. 212 15.2.2 Inference . 213 15.2.3 Computation. 213 15.2.4 Concretization. 214 15.2.5 Narration. 214 15.3 Fully Integrated Support. 215 15.4 Conclusions.215 A Metatheorems of 21 A.l A.2 A.3 A.4 A.5 A.6 2Ю Universal Instantiation. 216 Alpha-Conversion. 221 Substitution Rule. 223 Tautology Theorems. 228 Deduction Theorem . 234 Miscellaneous Metatheorems. 238 В Soundness of 21 240 C Henkin’s Theorem for 21 2^0
xii CONTENTS Bibliography 262 List of Figures 274 List of Tables 275 List of Theorems, Examples, Remarks, and Modules 276 Index 285 |
| adam_txt |
Contents Preface 1 Introduction Summary of the Contents. 2 Answers to Readers’ Questions 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3 Why Logic?. Why a Practical Logic?. . . Why Simple Type Theory?. Why not First-Order Logic?. Why not Set Theory?. Why not Dependent Type Theory?. Why Undefinedness?. Why Model Theory instead of Proof Theory?. Preliminary Concepts What is Mathematics?. Mathematical Values. 3.2.1 Sets. 3.2.2 Sequences. 3.2.3 Relations. 3.2.4 Functions. 3.2.5 Boolean Values and Predicates. 3.3 Binders. 3.4
Undefinedness. 3.5 Mathematical Structures. 3.6 Examples of Mathematical Structures. 3.7 Conclusions. 3.8 Exercises . 3.1 3.2 xiii 1 3 6 6 6 7 8 9 10 11 11 13 13 15 16 17 18 19 21 22 25 27 30 32 32 vii
CONTENTS viii 4 Syntax 5 4.1 Notation. 4.2 Symbols. 4.3 Types. 4.4 Expressions. 4.5 Bound and Free Variables. 4.6 Substitution. 4.7 Languages. 4.8 Conclusions. 4.9 Exercises . 42 42 44 44 45 Semantics 47 Interpretations. General Models. Finite General Models. Standard Models. 47 48 52 53 5.5 Satisfiability, Validity, and Semantic Consequence. 54 5.6 5.7 5.8 5.9 5.10 5.11 Isomorphic General Models. Expansion of a General
Model. Standard vs. General Semantics. Examples of Standard Models. Conclusions. Exercises . 56 57 58 59 62 62 5.1 5.2 5.3 5.4 6 Additional Notation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Boolean Operators. Binary Operators. Quantifiers . Definedness. Sets. Tuples. Functions. Miscellaneous Notation. Quasitypes and Dependent Quasitypes . Conclusions. Exercises . 7 Beta-Reduction and Substitution
7.1 Beta-Reduction. 7.2 Universal Instantiation. 64 64 67 68 69 70 70 70 70 72 75 75 77 77 g0
CONTENTS 7.3 7.4 7.5 7.6 Invalid Beta-Reduction. Alpha-Conversion. Conclusions. Exercises . ix 81 82 83 83 8 Proof Systems 84 8.1 Background. 84 8.2 A Proof System for Alonzo. 86 8.2.1 Axioms. 86 8.2.2 Rules of Inference . 88 8.2.3 Proofs. 88 8.3 Soundness. 89 8.4 Rugai General Models. 89 8.5 Completeness. 90 8.6 Conclusions. 93 8.7 Exercises . 93 9 Theories 94 9.1 Axiomatic Theories. 94 9.2 Theory
Extensions. 101 9.3 Conservative Theory Extensions . 104 9.4 Categorical Theories. Ш 9.5 Complete Theories. H$ 9.6 Fundamental Form of a Mathematical Problem. 120 9.7 Model Theory. 121 9.8 Conclusions. 122 9.9 Exercises . 12Ί 10 Sequences 12^ 10.1 Systems of Natural Numbers.12? 10.2 Notation for Sequences. 12$ 10.3 Conclusions . . 1$θ 10.4 Exercises . 1$θ 11 Developments ^1 11.1 Theory Developments. 1^1 11.2 Development of Natural Number Arithmetic. ■ · · 134 11.2.1 Basic Definitions and Theorems.I34 11.2.2 Commutative Semiring. 11.2.3 Weak Total
Order.136
CONTENTS x 11.2.4 Divides Lattice. ^^θ 11.3 Conclusions. 138 11.4 Exercises .138 12 Real Number Mathematics 139 12.1 Complete Ordered Fields . 140 12.2 Alternatives to the Construction of COF. 143 12.3 Development of Real Number Mathematics. 145 12.3.1 Some Basic Definitions and Theorems. 145 12.3.2 Naturals, Integers, and Rationals. 146 12.3.3 Iterated Sum and Product Operators. 147 12.3.4 Calculus. 148 12.3.5 Euclidean Space. 150 12.4 Skolem’s Paradox. 153 12.5 Conclusions. 154 12.6 Exercises . 154 13 Morphisms 156 13.1 A Motivating Example. 156 13.2 The Little Theories
Method. 158 13.3 Theory Morphisms. 159 13.3.1 Theory Translations. 159 13.3.2 Morphism Theorem . 164 13.3.3 Examples of TheoryMorphisms. 167 13.3.4 Faithful Theory Morphisms. 175 13.4 Development Morphisms. 177 13.4.1 Development Translations. 178 13.4.2 Transportations. 182 13.5 Mathematics Libraries. 187 13.5.1 Theory Graphs. 187 13.5.2 Development Graphs. 188 13.5.3 Realm Graphs . 188 13.6 Theory Graph Combinatore. 189 13.7 Conclusions. . 13.8 Exercises . 14 Alonzo Variants 14.1 Alonzo with Indefinite Description. 14.1.1
Introduction . 14.1.2 Syntax. 196 1gß լցք 1Q7
CONTENTS xi 14.1.3 Semantics. 197 14.1.4 Proof System. 197 14.1.5 Theorems. 198 14.2 Alonzo with Sorts . 199 14.2.1 Introduction . 199 14.2.2 Syntax. 199 14.2.3 Semantics. 201 14.2.4 Proof System. 203 14.2.5 Theorems . . 204 14.3 Alonzo with Quotation and Evaluation. 204 14.3.1 Introduction . 204 14.3.2 Syntax. 205 14.3.3 Semantics. 207 14.3.4 Proof System. 208 14.3.5 Theorems. 208 14.4 Conclusions. 209 15 Software Support 210 15.1 Basis
Support. 211 15.2 Advanced Support. 212 15.2.1 Organization. 212 15.2.2 Inference . 213 15.2.3 Computation. 213 15.2.4 Concretization. 214 15.2.5 Narration. 214 15.3 Fully Integrated Support. 215 15.4 Conclusions.215 A Metatheorems of 21 A.l A.2 A.3 A.4 A.5 A.6 2Ю Universal Instantiation. 216 Alpha-Conversion. 221 Substitution Rule. 223 Tautology Theorems. 228 Deduction Theorem . 234 Miscellaneous Metatheorems. 238 В Soundness of 21 240 C Henkin’s Theorem for 21 2^0
xii CONTENTS Bibliography 262 List of Figures 274 List of Tables 275 List of Theorems, Examples, Remarks, and Modules 276 Index 285 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-03T21:26:55Z |
| indexdate | 2024-12-16T13:02:44Z |
| institution | BVB |
| isbn | 9783031211140 9783031211119 |
| language | English |
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| publishDate | 2023 |
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| spelling | Farmer, William M. 19XX- Verfasser (DE-588)1281973637 aut Simple type theory a practical logic for expressing and reasoning about mathematical ideas by William M. Farmer Cham Birkhäuser [2023] Cham Springer International Publishing © 2023 XIV, 295 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Computer science foundations and applied logic Computer Science Logic and Foundations of Programming Mathematical Logic and Foundations Computational Complexity Formal Reasoning Set Theory Computer science Mathematical logic Computational complexity Reasoning Set theory Metamathematik (DE-588)4074759-1 gnd rswk-swf Logik (DE-588)4036202-4 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Metamathematik (DE-588)4074759-1 s Mathematik (DE-588)4037944-9 s Logik (DE-588)4036202-4 s DE-604 Erscheint auch als Online-Ausgabe 978-3-031-21112-6 Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034063321&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Farmer, William M. 19XX- Simple type theory a practical logic for expressing and reasoning about mathematical ideas Computer Science Logic and Foundations of Programming Mathematical Logic and Foundations Computational Complexity Formal Reasoning Set Theory Computer science Mathematical logic Computational complexity Reasoning Set theory Metamathematik (DE-588)4074759-1 gnd Logik (DE-588)4036202-4 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4074759-1 (DE-588)4036202-4 (DE-588)4037944-9 |
| title | Simple type theory a practical logic for expressing and reasoning about mathematical ideas |
| title_auth | Simple type theory a practical logic for expressing and reasoning about mathematical ideas |
| title_exact_search | Simple type theory a practical logic for expressing and reasoning about mathematical ideas |
| title_exact_search_txtP | Simple Type Theory A Practical Logic for Expressing and Reasoning About Mathematical Ideas |
| title_full | Simple type theory a practical logic for expressing and reasoning about mathematical ideas by William M. Farmer |
| title_fullStr | Simple type theory a practical logic for expressing and reasoning about mathematical ideas by William M. Farmer |
| title_full_unstemmed | Simple type theory a practical logic for expressing and reasoning about mathematical ideas by William M. Farmer |
| title_short | Simple type theory |
| title_sort | simple type theory a practical logic for expressing and reasoning about mathematical ideas |
| title_sub | a practical logic for expressing and reasoning about mathematical ideas |
| topic | Computer Science Logic and Foundations of Programming Mathematical Logic and Foundations Computational Complexity Formal Reasoning Set Theory Computer science Mathematical logic Computational complexity Reasoning Set theory Metamathematik (DE-588)4074759-1 gnd Logik (DE-588)4036202-4 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Computer Science Logic and Foundations of Programming Mathematical Logic and Foundations Computational Complexity Formal Reasoning Set Theory Computer science Mathematical logic Computational complexity Reasoning Set theory Metamathematik Logik Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034063321&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT farmerwilliamm simpletypetheoryapracticallogicforexpressingandreasoningaboutmathematicalideas |