Reddit mentions of Logic for Mathematicians

I just finished studying Gödel's incompleteness theorems. The tutor made us work from this book and it was not great. Full of errors and sometimes left proofs that were less than obvious as exercises. However, I picked up this book from the library and found it to be reasonably good (just be sure to look up the errata). The first half of it is quite basic if you have some previous experience with logic or mathematical logic. If you want something less formal on Gödel then read Nagel and Newman's book. One of the best informal books about formal logic I have ever read in terms of clarity of exposition.

I don't want to recommend any other books, as all the others I have used I have not used in their entirety. I would say that most logic books have mistakes or bad exposition. I think it occurs due to the people writing the books being so fluent in the topic that some proofs or points seem obvious to them but may not be to the reader. I recommend this website. There are a lot of good links there. Personally, I think the best way to learn logic is to have someone teach it to you since you will always need to be asking questions or having someone check your workings. And, like maths, just practice exercises and proofs until you can do them without thinking.

I know it may not be exactly what you were looking for but I hope it helps all the same. Good luck.