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

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

by Torkel Franzen
     
 

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… See more details below

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.

Product Details

ISBN-13:
9781568811758
Publisher:
Taylor & Francis
Publication date:
09/02/2004
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. 1Introduction1
Ch. 2Arithmetical preliminaries13
Ch. 3Primes and proofs31
Ch. 4The language of arithmetic43
Ch. 5The language of analysis59
Ch. 6Ordinals and inductive definitions71
Ch. 7Formal languages and the definition of truth85
Ch. 8Logic and theories97
Ch. 9Peano arithmetic and computability117
Ch. 10Elementary and classical analysis143
Ch. 11The recursion theorem and ordinal notations153
Ch. 12The incompleteness theorems171
Ch. 13Iterated consistency185
Ch. 14Iterated reflection199
Ch. 15Iterated iteration and inexhaustibility219

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