# Book:Elliott Mendelson/Introduction to Mathematical Logic/Sixth Edition

## Elliott Mendelson: Introduction to Mathematical Logic (6th Edition)

Published $\text {2015}$, Routledge

ISBN 978-1482237726

### Contents

Preface
Introduction
1. The Propositional Calculus
Propositional Connectives:Truth Tables
Tautologies
An Axiom System for the Propositional Calculus
Independence: Many-Valued Logics
Other Axiomatizations
2. First-Order Logic and Model Theory
Quantifiers
Parentheses
First-Order Languages and Their Interpretations: Satisfiability and Truth: Models
First Order Theories
Logical Axioms
Proper Axioms
Rules of Inference
Properties of First-Order Theories
Particularization Rule A4
Existential Rule E4
Rule C
Completeness Theorems
First-Order Theories with Equality
Definitions of New Function Letters and Individual Constants
Prenex Normal Forms
Isomorphism of Interpretations: Categoricity of Theories
Generalized First-Order Theories: Completeness and Decidability
Mathematical Applications
Elementary Equivalence: Elementary Extensions
Ultrapowers: Nonstandard Analysis
Reduced Direct Products
Nonstandard Analysis
Semantic Trees
Basic Principle of Semantic Trees
Quantification Theory Allowing Empty Domains
3. Formal Number Theory
An Axiom System
Number-Theoretic Functions and Relations
Primitive Recursive and Recursive Functions
Arithmetization: Godel Numbers
The Fixed-Point Theorem: Godel's Incompleteness Theorem
Recursive Undecidability: Church's Theorem
Nonstandard Models
4. Axiomatic Set Theory
An Axiom System
Ordinal Numbers
Equinumerosity: Finite and Denumerable Sets
Finite Sets
Hartogs' Theorem: Initial Ordinals - Ordinal Arithmetic
The Axiom of Choice: The Axiom of Regularity
Other Axiomatizations of Set Theory
Morse-Kelley (MK)
Zermelo-Fraenkel (ZF)
The Theory of Types (ST)
ST1 (Extensionality Axiom)
ST2 (Comprehension Axiom Scheme)
ST3 (Axiom of Infinity)
Quine's Theories NF and ML
NF1 (Extensionality)
NF2 (Comprehension)
Set Theory with Urelements
5. Computability
Algorithms: Turing Machines
Diagrams
Partial Recursive Functions: Unsolvable Problems
The Kleene-Mostowski Hierarchy: Recursively Enumerable Sets
Other Notions of Computability
Herbrand-Godel Computability
Markov Algorithms
Decision Problems
Appendix A: Second-Order Logic
Appendix B: First Steps in Modal Propositional Logic
Appendix C: A Consistency Proof for Formal Number Theory