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Let me explain my math background first. I recently finished "how to prove it" by velleman at home. That's it.

I stumbled upon "Introduction to Mathematical Logic, Sixth Edition" by Elliott Mendelson as a prerequisite for ordinal logic that alan turing helped pioneer.

Many amazon reviews on the fifth and sixth editions say it's not for self learners who are not familiar with logic already.

I'm starting to consider "An Introduction to Mathematical Logic" by Richard E. Hodel.

Can anyone give a good recommendation for my self-study material?

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    $\begingroup$ You would need to indicate what you mean by "beginning". Mendelson's book is fine for a learner who understands the methods of mathematics (we call this "mathematical maturity"), even if the learner doesn't know any mathematical logic. The book is well written and rigorous, and will demand the same effort and attention as any other advanced mathematics book. If you don't have exposure to advanced mathematics and proof, or if you are looking for a light read on the beach, then the book is probably not for you. As the the Amazon reviews, I wouldn't give them any weight at all. $\endgroup$ – Carl Mummert Oct 17 '15 at 12:22
  • $\begingroup$ @CarlMummert Did "how to prove it 2nd edition" by velleman prepare me for Mendelson's book? I studied velleman's book from the beginning to the end although I solved only 5-6 exercises per chapter. $\endgroup$ – crocket Oct 17 '15 at 12:24
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    $\begingroup$ If you have that background, but haven't done anything else in advanced math, you are in a position where you could work through Mendelson's book, but if it is the first advanced book you have worked through you will find that it takes time, that the information is presented in a compact way, and that you have to stop and think frequently. This is absolutely normal for advanced math books - it just takes some experience to acclimate yourself to the way that mathematics is presented. I would recommend also looking at Enderton's book to see if you find it more readable. $\endgroup$ – Carl Mummert Oct 17 '15 at 12:26
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    $\begingroup$ My experience with the fourth edition of Mendelson was that the part about logic is perfectly serviceable (with the caveats Carl mentions), the development of set theory is quite dense and can be heavy going, and for the love of Turing don't plan to learn computability theory from it. $\endgroup$ – Henning Makholm Oct 17 '15 at 12:35
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    $\begingroup$ Peter Smith has a very thorough summary of introductory logic books at logicmatters.net/students/tyl . For computability, Cutland's book is often used for undergraduate courses, but I feel it is too light. Enderton's computability book is has a very contemporary viewpoint, and I would recommend it. There is a nice, short book by Rebecca Weber that is slightly more advanced, but gives a better sense of how a computability theorist would think about the area. $\endgroup$ – Carl Mummert Oct 17 '15 at 12:57

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