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This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. The book is appropriate for use as a classroom text, for self-study, and as a reference on the state of modern logic.
Preface
Introduction
Propositional Logic and Other Fundamentals
The propositional language
Induction and recursion
Induction
Recursion
Propositional semantics
Propositional theories
General properties
Compactness
Decidability and effective enumerability
Other constructions
Notions of consistency
Ultraproducts
Topology and Boolean algebra
Topology
Boolean algebra
First-Order Logic
Syntax and semantics of first-order languages
Basic semantics
Substitution
Structures
Isomorphism and equivalence
Substructures
Products and chains
Theories
The language of equality
Dense linear orderings
Arithmetic
Changing languages
Interpretations
Completeness and Compactness
Countable compactness
Countable completeness
Other constructions
Notions of consistency
Ultraproducts
Boolean algebra
Uncountable languages and structures
Applications of compactness
Higher-order logic
Monadic second-order logic
Infinitary logic
Incompleteness and Undecidability
A first look
Recursive functions and relations
Recursively enumerable sets and relations
Gödel numbering
Definability in arithmetic I
Representability: First Incompleteness Theorem
Topics in Definability
Definability in arithmetic II
Indexing
Second Incompleteness Theorem
Church’s Thesis
Recursion equations
Abstract machines
Applications to other languages and theories
Set Theory
Zermelo-Fraenkel set theory
Mathematics in set theory I
Ordinal numbers: induction and recursion
Cardinal numbers
Models and independence
Mathematics in set theory II
The constructible universe
Generic extensions
Forcing
Large cardinals
Determinacy
Model Theory
Partial embeddings
Boolean algebras, ultrafilters and types
Countable models of countable theories
Uncountable models of countable theories
Morley’s Theorem
Abstract logics
Recursion Theory
Many-one degrees and r.e. sets
Turing reducibility
The jump operator
Upper bounds
Jumps of r.e. sets
Lower bounds
References
Item References
Symbol Index
Subject Index