Book:Elliott Mendelson/Introduction to Mathematical Logic/Sixth Edition
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Elliott Mendelson: Introduction to Mathematical Logic (6th Edition)
Published $\text {2015}$, Routledge
- ISBN 978-1482237726
Subject Matter
Contents
- Preface
- Introduction
- 1. The Propositional Calculus
- Propositional Connectives:Truth Tables
- Tautologies
- Adequate Sets of Connectives
- 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
- Additional Metatheorems and Derived Rules
- 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
- Quantifiers
- 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
- Answers to Selected Exercises
- Bibliography
- Notations
- Index