An Introduction to Gödel's TheoremsCambridge University Press, 26 juil. 2007 In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. |
Table des matières
Decidability and enumerability | 8 |
Axiomatized formal theories | 17 |
Capturing numerical properties | 28 |
The truths of arithmetic | 37 |
Sufficiently strong arithmetics | 43 |
Two formalized arithmetics | 51 |
What Q can prove | 58 |
Firstorder Peano Arithmetic | 71 |
Using the Diagonalization Lemma | 175 |
Secondorder arithmetics | 186 |
Incompleteness and Isaacsons conjecture | 199 |
Gödels Second Theorem for PA | 212 |
The derivability conditions | 222 |
Deriving the derivability conditions | 232 |
Reflections | 240 |
About the Second Theorem | 252 |
Primitive recursive functions | 83 |
Capturing p r functions | 99 |
Q is p r adequate | 106 |
A very little about Principia | 118 |
The arithmetization of syntax | 124 |
PA is incomplete | 138 |
Gödels First Theorem | 147 |
About the First Theorem | 153 |
Strengthening the First Theorem | 162 |
The Diagonalization Lemma | 169 |
μRecursive functions | 265 |
Undecidability and incompleteness | 277 |
Turing machines | 287 |
Turing machines and recursiveness | 298 |
Halting problems | 305 |
The ChurchTuring Thesis | 315 |
Looking back | 342 |
| 356 | |
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Expressions et termes fréquents
algorithm argument assuming assumption axioms basic arithmetic bounded quantifiers canonical Gödel sentence captures chapter characteristic function Church-Turing Thesis Church's Thesis claim computable functions consistent construction dataspace defined derivability conditions Diagonalization Lemma effectively computable effectively decidable effectively enumerable equivalent example fact finite first-order logic function f given Gödel numbers Gödelian Goldbach type Goodstein's theorem halting problem halts Hence Here's idea Incompleteness Theorem input interpretation intuitive language Löb's Theorem loops mathematics natural numbers negation complete nice theory numerical property one-place open wff p.r. adequate p.r. definition p.r. function PA's predicate primitive recursive primitive recursive function Proof sketch provable prove quantifiers recursively solvable result scanned cell Second Theorem second-order second-order arithmetic Section semantic sequence successor suppose symbols theory of arithmetic total functions trivial true truths Turing machine Turing program undecidable unprovable w-consistent zero µ-recursive function