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Mathematical Logic A Course with Exercises - Recursion Theory, Godel's Theorems, Set Theory, Model Theory

ISBN-10: 0198500505

ISBN-13: 9780198500506

Edition: 2001

Authors: Ren� Cori, Daniel Lascar, Donald Pelletier

List price: $92.00
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Description:

Logic forms the basis of mathematics, and is hence a fundamental part of any mathematics course. It is a major element in theoretical computer science and has undergone a huge revival with the every- growing importance of computer science. This text is based on a course to undergraduates and provides a clear and accessible introduction to mathematical logic. The concept of model provides the underlying theme, giving the text a theoretical coherence whilst still covering a wide area of logic. The foundations having been laid in Part I, this book starts with recursion theory, a topic essential for the complete scientist. Then follows Godel's incompleteness theorems and axiomatic set theory. Chapter 8 provides an introduction to model theory. There are examples throughout each section, and varied selection of exercises at the end. Answers to the exercises are given in the appendix.
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Book details

List price: $92.00
Copyright year: 2001
Publisher: Oxford University Press, Incorporated
Publication date: 6/21/2001
Binding: Paperback
Pages: 352
Size: 6.50" wide x 9.50" long x 1.00" tall
Weight: 1.100
Language: English

Contents of Part I
Notes from the translator
Notes to the reader
Introduction
Recursion theory
Primitive recursive functions and sets
Some initial definitions
Examples and closure properties
Coding of sequences
Recursive functions
Ackerman's function
The [mu]-operator and the partial recursive functions
Turing machines
Description of Turing machines
T-computable functions
T-computable partial functions are recursive
Universal Turing machines
Recursively enumerable sets
Recursive and recursively enumerable sets
The halting problem
The smn theorem
The fixed point theorems
Exercises for Chapter 5
Formalization of arithmetic, Godel's theorems
Peano's axioms
The axioms
The ordering on the integers
Representable functions
Arithmetization of syntax
The coding of formulas
The coding of proofs
Incompleteness and undecidability theorems
Undecidability of arithmetic and predicate calculus
Godel's incompleteness theorems
Exercises for Chapter 6
Set theory
The theories Z and ZF
The axioms
Ordered pairs, relations, and maps
Ordinal numbers and integers
Well-ordered sets
The ordinals
Operations on ordinal numbers
The integers
Inductive proofs and definitions
Induction
The axiom of choice
Cardinality
Cardinal equivalence classes
Operations on cardinal equivalence classes
The finite cardinals
Countable sets
The cardinal numbers
The axiom of foundation and the reflection schemes
The axiom of foundation
Some relative consistency results
Inaccessible cardinals
The reflection scheme
Exercises for Chapter 7
Some model theory
Elementary substructures and extensions
Elementary substructures
The Tarski-Vaught test
Construction of elementary extensions
Elementary maps
The method of diagrams
The interpolation and definability theorems
Reduced products and ultraproducts
Preservation theorems
Preservation by substructures
Preservation by unions of chains
Preservation by reduced products
[characters not reproducible]-categorical theories
The omitting types theorem
[characters not reproducible]-categorical structures
Exercises for Chapter 8
Solutions to the exercises of Part II
Bibliography
Index