An Introduction to Metalogic

ISBN-13: 9781554811717 / Angielski / Miękka / 2014 / 344 str.

Aladdin M. Yaqub

Opis

An Introduction to Metalogic is a uniquely accessible introduction to the metatheory of first-order predicate logic. No background knowledge of logic is presupposed, as the book is entirely self-contained and clearly defines all of the technical terms it employs. Yaqub begins with an introduction to predicate logic, and ends with detailed outlines of the proofs of the incompleteness, undecidability and indefinability theorems. Many other topics are covered: expressive completeness, the basics of set theory, infinite cardinalities, Cantor's Theorems, the Soundness and Completeness Theorems, Turing machines, the Halting Problem, predicate logic theories and their properties, elementary equivalence, isomorphism, Peano Arithmetic, second-order predicate logic and more.

Spis treści

• Introduction
• Chapter One: First-Order Predicate Logic
• 1. The Syntax of PL
• 2. The Semantics of PL
• 3. Logical Concepts in PL
• 4. PL Proof Theory
• 5. Exercises
• Chapter Two: Resources of the Metatheory
• 1. Linguistic and Logical Resources
• 2. Arithmetical Resources
• 3. Set-Theoretic Resources
• 4. An Economical Version of PL
• 5. Exercises
• Chapter Three: The Soundness and Completeness Theorems
• 1. The Soundness Theorem
• 2. The Completeness Theorem
• 3. The Compactness Theorem
• 4. PL Interpretations and PL Sets
• 5. The Löwenheim-Skolem Theorem
• 6. Exercises
• Chapter Four: Computability
• 1. Effective Procedures and Computable Functions
• 2. Turing Computability
• 3. The Halting Problem
• 4. Partial Recursive Functions
• 5. Exercises
• Chapter Five: The Incompleteness Theorems
• 1. Peano Arithmetic
• 2. Representability in Peano Arithmetic
• 3. The Arithmetization of the Metatheory
• 4. Diagonalization and the First Incompleteness Theorem
• 5. Consequences of Diagonalization and Incompleteness
• 6. The Incompleteness of Second-Order Predicate Logic
• 7. Gödel's Second Incompleteness Theorem
• 8. Exercises