Mathematical Logic and Formalized Theories

Mathematical Logic and Formalized Theories

A Survey of Basic Concepts and Results

2nd Edition - January 1, 1971

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  • Author: Robert Rogers
  • eBook ISBN: 9781483257976

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Description

Mathematical Logic and Formalized Theories: A Survey of Basic Concepts and Results focuses on basic concepts and results of mathematical logic and the study of formalized theories. The manuscript first elaborates on sentential logic and first-order predicate logic. Discussions focus on first-order predicate logic with identity and operation symbols, first-order predicate logic with identity, completeness theorems, elementary theories, deduction theorem, interpretations, truth, and validity, sentential connectives, and tautologies. The text then tackles second-order predicate logic, as well as second-order theories, theory of definition, and second-order predicate logic F2. The publication takes a look at natural and real numbers, incompleteness, and the axiomatic set theory. Topics include paradoxes, recursive functions and relations, Gödel's first incompleteness theorem, axiom of choice, metamathematics of R and elementary algebra, and metamathematics of N. The book is a valuable reference for mathematicians and researchers interested in mathematical logic and formalized theories.

Table of Contents


  • I The Sentential Logic

    1.1. Introduction

    1.2. Sentential Connectives

    1.3. The Sentential Logic P. Symbols and Formulas

    1.4. Tautologies

    1.5. Axiom Schemata of P. Rules of Inference and Theorems

    1.6. Metamathematical Properties of P

    II The First-Order Predicate Logic: I

    2.1. The First-Order Predicate Logic F1. Symbols, Quantifiers and Formulas

    2.2. Interpretations. Truth and Validity

    2.3. Axiom Schemata of F1. Rules of Inference and Theorems. Consistency of F1

    2.4. The Deduction Theorem

    III The First-Order Predicate Logic: II

    3.1. Elementary Theories

    3.2. Completeness Theorems

    3.3. Further Corollaries. Decision Problem

    3.4. The First-Order Predicate Logic with Identity

    3.5. The First-Order Predicate Logic with Identity and Operation Symbols

    IV The Second-Order Predicate Logic. Theory of Definition

    4.1. Introduction

    4.2. The Second-Order Predicate Logic F2

    4.3. Second-Order Theories

    4.4. Theory of Definition

    V The Natural Numbers

    5.1. Introduction

    5.2. Elementary Arithmetic: The Theory N

    5.3. The Metamathematics of N

    5.4. Second-Order Arithmetic: The Theory N2

    5.5. The Metamathematics of N2

    VI The Real Numbers

    6.1. The Theory R

    6.2. The Metamathematics of R and of Elementary Algebra

    6.3. Second-Order Real Number Theory: The Theory R2

    6.4. The Metamathematics of R2

    VII Axiomatic Set Theory

    7.1. Paradoxes

    7.2. The Zermelo-Fraenkel Axioms

    7.3. The Axiom of Choice

    7.4. The Metamathematics of ZF

    7.5. Strengthened Forms of ZF

    VIII Incompleteness. Undecidability

    8.1. Introduction

    8.2. Recursive Functions and Relations. Representability

    8.3. Arithmetization

    8.4. Gödel's First Incompleteness Theorem

    8.5. Gödel's Second Incompleteness Theorem

    8.6. Tarski's Theorem

    8.7. Decision Problem. Church's Thesis. Recursively Enumerable Sets

    8.8. Undecidability

    Author Index

    Subject Index

Product details

  • No. of pages: 248
  • Language: English
  • Copyright: © North Holland 1971
  • Published: January 1, 1971
  • Imprint: North Holland
  • eBook ISBN: 9781483257976

About the Author

Robert Rogers

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