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Is there a modern book on Gödel's incompleteness theorems that goes into each and every technical aspect of the proof of them (a classical one, if such exists)? I'm not interested in popular literature that constantly draws analogies with computers, printers, etc. I want the real thing.

P.S. I also started reading Gödel's 1931 original paper, but thought that since then the proof could have become more elegant and simple.

Martin Sleziak
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user132181
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    Does the list at http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Books_about_the_theorems suffice? – lhf Apr 14 '14 at 14:01
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    It would be cool if you could actually point to one book that stands out in general and matches my criteria. There are far more books in that list that I could process or have access to. – user132181 Apr 14 '14 at 14:06
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    Some of the ones in @lhf's link (e.g. Hofstadter) are probably too popular for the OP. I believe Peter Smith's book is technically solid. – hmakholm left over Monica Apr 14 '14 at 14:07
  • @CarlMummert, I do too, though searching didn't help. – user132181 Apr 14 '14 at 14:12
  • @user132181: does http://math.stackexchange.com/questions/52320/mathematician-non-logician-seeks-reference-for-godels-incompleteness-theorems address your needs? – Carl Mummert Apr 14 '14 at 14:12
  • @CarlMummert, no, quite the opposite. – user132181 Apr 14 '14 at 14:14
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    @user132181: in that case, the question is not a duplicate! I can write an answer, but I will wait a little while to see if a lower-rep user would prefer to write one first. I'll check back in a a few hours. – Carl Mummert Apr 14 '14 at 14:17
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    Peter Smith's Introduction to Godel's Theorems is a readable introduction that includes the sort of technical details you seem to be interested in; however if you want "each and every technical aspect" it may not be sufficient. At the very least, its a good place to start. – goblin GONE Apr 14 '14 at 14:30
  • I think that Peter Smyth's book is pretty good, http://www.logicmatters.net/igt/ – hmmmm Apr 14 '14 at 14:34
  • @CarlMummert, has Mauro already given the answer you wanted to give? – user132181 Apr 14 '14 at 17:28
  • Enderton's "Introduction to Mathematical Logic" contains detailed proofs of the incompleteness theorems, and is a stellar book in general. –  Apr 14 '14 at 19:13
  • If I had to pick just one I'd pick Smullyan's. – MikeC Apr 15 '14 at 14:50

2 Answers2

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There are several senses of "complete":

  • If you want a complete discussion of the incompleteness theorems and their related computability and philosophical concepts, the best modern reference is Peter Smith's book An Introduction to Gödel's Theorems.

  • If you want a complete technical proof of the theorems, but with little discussion of computability and without philosophical asides, then Smorynski's article "The incompleteness theorems" in the Handbook of Mathematical Logic is an exceptional reference. This article includes quite general statements of the theorems and results on formalizing the incompleteness theorems into systems such as PRA. This paper was also mentioned in this answer. The paper is written as a reference paper in a research-level handbook, so the ideal reader needs to be prepared for exposition at that level.

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Carl Mummert
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Besides Peter Smith's book (An Introduction to Gödel's Theorems, 2nd ed 2013, Cambridge UP), I suggest (see Wiki and SEP bibliographies) :

Raymond Smullyan, 1991, Gödel's Incompleteness Theorems, Oxford Univ.Press

Roman Murawski, 1999, Recursive Functions and Metamathematics: Problems of Completeness and Decidability, Gödel's Theorems, Kluwer A.P.

Torkel Franzén, 2004, Inexhaustibility: A Non-Exhaustive Treatment, Lecture Notes in Logic 16, A.K.Peters

Torkel Franzén, 2005, Gödel's Theorem: An Incomplete Guide to its Use and Abuse, A.K.Peters.

See also the textbook :

George Tourlakis, 2003, Lectures in Logic and Set Theory. Volume 1 : Mathematical Logic, Cambridge UP;

all the 2nd part of the book (from page 155 until 315) is dedicated to a detailed exposition of 1st and 2nd (and this is not easy to find in textbooks) Gödel's Incompleteness Theorems.

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Mauro ALLEGRANZA
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