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I am a student who is working Lie Theory. I want to start read theory of logics. I just need some reference and I have few questions regarding this,

i) will studying theory of logics will improve my theorem proving ability in 'other branch' of mathematics?

ii) is there any results in mathematics, (I am sure there are) (say in algebra particularly in group theory) proof of which is made easy through the approaches and techniques in theory of logics?

iii) where to start reading logic theory and which are good text books in logic theory for the beginners.

I have read Paul R.Halmos 's Naive Set Theory book completely. But I haven't done all the exercises. This is the amount of knowledge I have regarding theory of logics.

I hope I am not confusing between theory of sets and theory of logics.

Thanks for your valuable suggestions.

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Yes, there are for example results in algebra that are proved by model-theoretic means (and model theory is usually taken to be a core part of mathematical logic). A classic early example is touched on here http://bit.ly/JKyuuZ And for lots of references to different areas where model theory has got applied more recently, see this MO post and the answers: http://bit.ly/4IXUHa

The logical resources needed for such applications are pretty sophisticated, but if you want to make a start learning some logic I'm inclined to think that (if you are a good mathematics student) Peter Hinman's Fundamentals of Mathematical Logic is perhaps the go-to book if you want a single textbook to get you into the field. It is very lucid. True, it is a Very Big Book, but that's because it covers a lot at an accessible pace -- and the preface explains very well how to steer a course through the book to suit particular interests.

For an extensive annotated reading Guide to logic more generally, looking at lot of textbooks at different levels, you can download the PDF http://bit.ly/18t5CUD (which doesn't yet given an account of Hinman's book -- a major shortcoming that will be corrected in the next version). The Guide has a section on model theory in particular.

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    $\begingroup$ I am eager to know few such examples from algebra. can you tell some of them or any reference where I can find such examples? thanks. $\endgroup$ – GA316 Nov 9 '13 at 10:25
  • $\begingroup$ I've added a couple of references to the answer. Sorry, I should have done that from the beginning! $\endgroup$ – Peter Smith Nov 9 '13 at 11:39
  • $\begingroup$ thanks for your interest in sharing knowledge and experience. $\endgroup$ – GA316 Nov 9 '13 at 16:40
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    $\begingroup$ Why the conversion of informative URLs into bit.ly minis? I don't know about anyone else, but I find it much more readable either just have them inlined ('a classic early example is touched on here', with here hotlinked) or the full URLs given and used as an object as part of the sentence ('see the post at mathoverflow.net/questions/7018') . The bit.ly URLs just 'don't fit' in some sense, for me, and I actually find them a little disruptive to read around/through... $\endgroup$ – Steven Stadnicki Nov 10 '13 at 17:06
  • $\begingroup$ @StevenStadnicki Dunno what got into me! ... Not my usual habit. $\endgroup$ – Peter Smith Nov 10 '13 at 18:07

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