Sentences Undecidable in Formalized Arithmetic
An Exposition of the Theory of Kurt Godel
Published on 14. May 2014
127 pages
978-0-08-095730-2 (ISBN)
Description
Sentences Undecidable in Formalized Arithmetic
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Lev D. Beklemishev
Provability, Computability and Reflection: Volume 10
Book
04/2000
Elsevier
€210.46
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Content
- Front Cover
- Sentences Undecidable in Formalized Arithmetic: An Exposition of the Theory of Kurt Gödel
- Copyright Page
- Preface
- Contents
- Introduction
- Chapter I. Auxiliary notions and theorems of arithmetic
- 1. Mapping of integers onto pairs and triples of integers
- 2. Mapping of integers onto finite sequences of integers
- 3. Logical symbols
- 4. Relations and functions
- 5. Inductive definitions
- Chapter II. The system (S) and its syntax
- 1. Preliminary description of the system
- 2. Expressions and integers
- 3. Properties of functional forms and matrix forms
- 4. Explicit definitions of sets Jf, IRf, and of functions Oc, Ind, and S
- 5. Axioms of (S)
- 6. Formal proofs
- Chapter III. Arithmetical theorems provable in (S)
- 1. Abbreviations
- 2. The propositional calculus
- 3. The functional calculus
- 4. Theory of identity
- 5. Induction
- 6. Addition and multiplication of digits
- 7. Theorems on inequalities
- 8.Matrices satisfied by exactly one digit
- Chapter IV. Semantics of (S)
- 1. Representability of infinite sequences
- 2. Values of functional forms and the notion of satisfaction for matrix forms
- 3. Inductive definition of Val(f, m) and GtSf(F, m)
- 4. Explicit definition of Val and GtSf
- 5. Class Lr
- Chapter V. Recursivity and definability of functions and relations
- 1. The notion of A-definability
- 2. General properties of 8-definable functions and relations
- 3. Recursivity of functions and relations corresponding to the syntactical notions
- 4. A theorem concerning the min-operator
- 5. Recursively enumerable sets
- Chapter VI. Proofs of incompleteness theorems
- 1. A-undefinability of the set LA
- 2. Proofs of the incompleteness theorem
- 3. Generalizations
- 4. Discussion of the incompleteness theorems
- Appendix. Some further results of gödel's theory
- 1. Undecidability of the sentence expressing the consistency of (S)
- 2. Problem of truth of undecidable sentences
- 3. Sentences provable simultaneously in (S) and (S1)
- Bibliography
