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I want to study Godel's incompleteness theorems and I look for a text which provide mathematical logic with a nice way to make me able to study Godel's incompleteness theorems

I didn't study mathematical logic before. some people recommended Rene Cori and Daniel lascar text. what do you think ?

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I almost feel like preempting Peter Smith and recommending his book, An Introduction to Gödel's Theorems, whose second edition is forthcoming. But I won't. – arjafi Jan 9 '13 at 17:44
up vote 11 down vote accepted

Well, how can I resist? My Introduction to Gödel's Theorems (CUP, 2007; second edition forthcoming in March 2013) was written precisely for those with little prior background in mathematical logic, and aims to be very accessible while actually giving the main proofs in quite a bit of detail.

Other options that range a bit more widely into logic and the theory of computation include George Boolos and Richard C. Jeffrey, Computability and Logic (CUP, 3rd edn 1989: there are expanded later editions with John Burgess as a third author, but these are arguably less elegantly done). Richard L. Epstein and Walter Carnielli’s very nice Computability also discusses computation in general before covering the incompleteness theorems; this is attractively written with a lot of interesting historical asides.

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I realise that my comment above may be taken as a slight directed at you: be assured that no slight was intended. I greatly appreciate that you share your insight on many logical matters here on math.SE. – arjafi Jan 10 '13 at 13:29
Oh no worries, @ArthurFischer, it made me smile and I took it in the spirit intended! – Peter Smith Jan 10 '13 at 13:31
thank you prof Peter Smith , I read the content , and found it Great ! it cover also turing machines and many subjects , it's a great text I think " I know little about turing general machine from prof Roger Penrose book " mind , computer and the laws of physics " " I read that your intro text is for philosophy students and first semester undergrad students in maths , – Maths Lover Jan 11 '13 at 2:43
is there other text can I use as a intro text to mathematical logic and enable me to study godel's theorems in more detail later ? what do you think about Rene Cori and Daniel lascar text ? is it good ? do you recommend it also to study mathematical logic ? or other text ? – Maths Lover Jan 11 '13 at 2:47
@MathsLover There's a lot of reading advice at – Peter Smith Jan 11 '13 at 10:29

I would like to recommend A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity. I am in a similar position as yourself, though probably a week ahead, as in without a prior background in symbolic Logic but curious to learn Godel's incompleteness theorem's. I started with this book last week. I am finding it to be a rigrous and pleasurable exposition.

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This is not a complete answer, but: If you are just getting started and you are particularly interested in Gödel's Theorems, then I can recommend Gödel's Proof by Nagel and Newman as a gentle introduction. The nice thing about this book is that it is relatively short and not too technical.

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in the truth , I look for a technical way , that means , i look for a good introduction to mathematical logic which make me able to study godel's theorems :) , – Maths Lover Jan 11 '13 at 2:23
@MathsLover: In that case, the book will not be the best, but since you wrote that you have not studied mathematical logic before, it might (might not) be helpful to start with a gentle introduction. The book actually outlines the proof, so when you get another book with the details, you will already know a bit about the ideas behind the arguments. It is very easy to get lost in technicalities. – Thomas Jan 11 '13 at 15:23

Boolos's "explanation" in words of one syllable published in Mind fits on the one page they let you see of the article (without a subscription). It's cute, but not what you're looking for. Still, it's worth a look, if only for amusement. You could think of it as a test of how well you are understanding the proof(s) you are reading.

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I learned logic from "Mathematical Logic," by Kleene. It starts from the basics and then goes into Godel's theorems. It's a very solid book, with plenty of examples.

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