Languages with Expressions of Infinite Length (eBook)

Lev D. Beklemishev (Autor)

eBook Download: PDF

2000 | 1. Auflage
182 Seiten
Elsevier Science (Verlag)
978-0-08-095756-2 (ISBN)

Infos

Provability, Computability and Reflection

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Languages with Expressions of Infinite Length

Inhalt

Front Cover 1
Languages with Expressions of Infinite Length 4
Copyright Page 5
Table of Contents 8
Foreward on Set Theory 12
Foreward on Alebra 16
Chapter 1. Introduction 22
1.1 An informal look at infinitary predicate languages 22
1.2 An informal look at formal systems based on infinitary predicate languages 26
Chapter 2. Infinitary Concatenation 30
2.1 Properties of infinitary ordinal addition 30
2.2 Properties of infinitary concatenation of sequences 31
2.3 Lemma for replacement of non-overlapping parts within sequences 34
Chapter 3. Algebras of Terms of Infinite Length 38
3.1 The o-algebras of terms 38
3.2 Algebras for the interpretation of terms 39
3.3 Recursion principle 41
3.4 Replacement of equivalent parts 44
3.5 Substitution 47
Chapter 4. Infinitary Propositional Languages 51
4.1 The a-propositional languages 51
4.2 Algebras of equivalence classes of formulas modulo a semantically consistent set 54
4.3 Interpreting a-propositional languages in -> a-complete Boolean algebras
Chapter 5. Infinitary Propositional Logic 60
5.1 Description of the formal systems for a-propositional languages 61
5.2 Development of the formal systems for a-propositional languages 64
5.3 Completeness of the basic formal systems when a=.1 68
5.4 Completeness of the basic formal systems with Chang's distributive laws 70
5.5 The basic formal systems with ordinary distributive laws 72
Chapter 6. Representation Theory for Boolean Algebras 76
6.1 Boolean algebraic equations corresponding to a-propositional schemes 76
6.2 Formally consistent sets of formulas 77
6.3 Algebras of equivalence classes of formulas modulo a formally consistent set 78
6.4 Metamathematical proofs of some Boolean representation theorems 80
6.5 Algebraic proof of a completeness theorem 83
Chapter 7. Non-Deducibility in Infinitary Propositional Logic 86
7.1 Summary of results and open problems 86
7.2 Examples of complete .-representable, not .+-representable Boolean algebras 89
Chapter 8. Systems of Formulas of Infinite Length 92
8.1 The (a,ß,.,p)-systems of formulas 93
8.2 Systems for the interpretation of formulas 94
8.3 Recursion principle 97
8.4 Replacement of equivalent parts 102
Chapter 9. Substitution 105
9.1 Free and bound occurrences of variables in formulas 105
9.2 Substitution for free variables 109
9.3 Substitution for bound variables 115
9.4 Function notation for substitution 120
Chapter 10. Infinitary Predicate Languages 122
10.1 The (a,ß,.,p)-predicate languages 122
10.2 Semantic consistency, Hanf's example 125
10.3 A criterion for satisfiability 129
10.4 Algebras of equivalence classes of formulas modulo a semantically consistent set 137
Chapter 11. Infinitary Predicate Logic 140
11.1 Description of the formal systems for (a,ß,.,p)-predicate languages 140
11.2 Development of the formal systems for (a,ß,.,p)-predicate languages 143
11.3 Completeness of the basic formal systems with Chang’s distributive laws and the rule of dependent choices for certain a and ß 148
11.4 Completeness of the basic formal systems when a = .1, ß = . 154
11.5 The reduction to the rule of independent choices when ß = . 155
Chapter 12. Non-Deducibility in Infinitary Predicate Logic 158
12.1 Summary of results and open problems 159
12.2 Boolean algebraic methods 160
12.3 General models 162
12.4 Non-deducibility of the axiom of dependent choices in systems with distributive laws and rules of independent choices 165
Chapter 13. The Definability of the Infinitary Formal Systems 173
13.1 The metalanguage for (a,ß,.,p)-formal systems 173
13.2 The definability of fundamental notions of set theory 175
13.3 The definability of the formal systems 178
Chapter 14. Incompleteness in Infinitary Predicate Logic 187
14.1 The basic undefinability argument 189
14.2 The incompleteness of definable formal systems when a non- limit, ß = a 190
References 196
Index of Symbols 200
Subject Index 202

Produkt-Details

• Erscheint lt. Verlag: 1.4.2000
• Sprache: englisch
• Themenwelt: Informatik ► Software Entwicklung ► User Interfaces (HCI)
• Themenwelt: Informatik ► Theorie / Studium ► Algorithmen
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-095756-0 / 0080957560
• ISBN-13: 978-0-08-095756-2 / 9780080957562