An Introduction to Gödel's Theorems
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 - extensively rewritten for its second edition - 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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I have read the first edition of this title and have found it to be one of the best, if not _the_ best introductions to the topic (possibly second only to Boolos et. al. "Computability and Logic". I am looking forward to the 2nd Edition.
Please note that at present (October 2013) the 'View Ebook' link on the page for the 2nd edition takes you to the purchase page for the *1st* edition ebook. Caveat Emptor.
Google, when this was pointed out to them, suggested I contact Peter Smith to get the issue corrected.
Functions and enumerations
Effectively axiomatized theories
Capturing numerical properties
The truths of arithmetic
What Q can prove
Firstorder Peano Arithmetic
Incompleteness and lsaacsons Thesis
Godels Second Theorem For PA
On the unprovability of consistency
Generalizing the Second Theorem
Lobs Theorem and other matters
Deriving the derivability conditions
The Second Theorem Hilbert minds and machines
l5 LA can express every p r function
A very little about Principia
The arithmetization of syntax
2O Arithmetization in more detail
PA is incomplete
Godels First Theorem
About the First Theorem
The Diagonalization Lemma
Broadening the scope