Inexhaustibility: A Non-Exhaustive Treatment: Lecture Notes in Logic 16

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Overview

Gödel's Incompleteness Theorems are among the most significant results in the foundation of mathematics. These results have a positive consequence: any system of axioms for mathematics that we recognize as correct can be properly extended by adding as a new axiom a formal statement expressing that the original system is consistent. This suggests that our mathematical knowledge is inexhaustible, an essentially philosophical topic to which this book is devoted. Basic material in predicate logic, set theory and recursion theory is presented, leading to a proof of incompleteness theorems. The inexhaustibility of mathematical knowledge is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman. All concepts and results necessary to understand the arguments are introduced as needed, making the presentation self-contained and thorough.
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Editorial Reviews

From the Publisher
"In this book the author discusses Gödel's famous incompleteness theorems. Special emphasis is put on the consequences of the inexhaustibility of our mathematical knowledge in any one formal axiomatic theory ... The book can be considered as a more technical companion to the author's more philosophical book [Gödel's Theorem, A K Peters, Wellesley, MA, 2005]." -Mathematiacl Reviews, November 2007
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Product Details

  • ISBN-13: 9781568811758
  • Publisher: Taylor & Francis
  • Publication date: 9/2/2004
  • Series: Lecture Notes in Logic Series
  • Pages: 263
  • Product dimensions: 6.00 (w) x 8.90 (h) x 0.60 (d)

Table of Contents

Ch. 1 Introduction 1
Ch. 2 Arithmetical preliminaries 13
Ch. 3 Primes and proofs 31
Ch. 4 The language of arithmetic 43
Ch. 5 The language of analysis 59
Ch. 6 Ordinals and inductive definitions 71
Ch. 7 Formal languages and the definition of truth 85
Ch. 8 Logic and theories 97
Ch. 9 Peano arithmetic and computability 117
Ch. 10 Elementary and classical analysis 143
Ch. 11 The recursion theorem and ordinal notations 153
Ch. 12 The incompleteness theorems 171
Ch. 13 Iterated consistency 185
Ch. 14 Iterated reflection 199
Ch. 15 Iterated iteration and inexhaustibility 219
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