An Introduction to Gödel's Theorems

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Cambridge University Press, 26 lug 2007
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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.
 

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Sommario

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
Firstorder Pea no Arithmetic
71
Primitive recursive functions
83
Using the Diagonalization Lemma
175
Secondorder arithmetics
186
Incompleteness and Isaacsons conjecture
199
Godels Second Theorem for PA
212
The derivability conditions
222
Deriving the derivability conditions
232
Reflections
240
About the Second Theorem
252

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
Godels First Theorem
147
About the First Theorem
153
Strengthening the First Theorem
162
2O The Diagonalization Lemma
169
pRecursive functions
265
3O Undecidability and incompleteness
277
Turing machines
287
Turing machines and recursiveness
298
Halting problems
305
The ChurchTuring Thesis
315
Looking back
342
Index
356
Copyright

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Informazioni sull'autore (2007)

Peter Smith is Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is a former editor of the journal Analysis.

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