Truth and the Unprovability of Consistency

It might be thought that we could argue for the consistency of a mathematical theory T within T, by giving an inductive argument that all theorems of T are true and inferring consistency. By Gödel’s second incompleteness theorem any such argument must break down, but just how it breaks down depends...

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Bibliographic Details
Published inMind Vol. 115; no. 459; pp. 567 - 606
Main Author Field, Hartry
Format Journal Article
LanguageEnglish
Published Oxford Oxford University Press 01.07.2006
Oxford Publishing Limited (England)
Subjects
Excluded middle
Inference
Logic
Logical theorems
Mathematical analysis
Mathematical set theory
Modus ponens
Paradoxes
Philosophy
Predicates
Priests
Truth
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Summary:It might be thought that we could argue for the consistency of a mathematical theory T within T, by giving an inductive argument that all theorems of T are true and inferring consistency. By Gödel’s second incompleteness theorem any such argument must break down, but just how it breaks down depends on the kind of theory of truth that is built into T. The paper surveys the possibilities, and suggests that some theories of truth give far more intuitive diagnoses of the breakdown than do others. The paper concludes with some morals about the nature of validity and about a possible alternative to the idea that mathematical theories are indefinitely extensible.
Bibliography:local:fzl567
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ISSN:0026-4423
1460-2113
DOI:10.1093/mind/fzl567