8
$\begingroup$

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.

$\endgroup$
12
  • 1
    $\begingroup$ Does the list at en.wikipedia.org/wiki/… suffice? $\endgroup$
    – lhf
    Apr 14, 2014 at 14:01
  • 2
    $\begingroup$ 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. $\endgroup$
    – user132181
    Apr 14, 2014 at 14:06
  • 3
    $\begingroup$ 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. $\endgroup$ Apr 14, 2014 at 14:07
  • 1
    $\begingroup$ @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. $\endgroup$ Apr 14, 2014 at 14:17
  • 1
    $\begingroup$ 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. $\endgroup$ Apr 14, 2014 at 14:30

2 Answers 2

8
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ Smith's book is great, though lacks it being completely formal (which is not a bad thing, but not what I'm looking for). Smullyan's book I don't like at all. The title of Murawski's one sounds great (showing that Gödel's theorems are really about recursive functions, and not arithmetic in particular), I really need to check that book out. Franzén's 2005 book is very informal and aimed at a layperson. The remaining two I haven't checked out yet, but I will. $\endgroup$
    – user132181
    Apr 14, 2014 at 17:24
  • $\begingroup$ Didn't Smullyan's book give one reader the impression that the incompleteness theorems only apply to Peano arithmetic? $\endgroup$ Jun 24, 2022 at 11:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .