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This is THE book to learn real recursion theory from. It isn't as good a reference as Odefreddi (though soare has more on the r.e. degrees) but it actually explains what is happening and offers insight into the subject. This isn't a book for the non-mathematician who just wants to know what all this stuff about Turing machines is about. This is a full course that takes one from the beginnings of the subject to a point where you are ready to read (and produce) recent papers. I personally wish it had a little bit more on forcing constructions and other non-re constructions (this part is frustratingly short) but the book is more coherent and reasonably sized for their omission. The approach taken is more modern than that of Rogers and much preferable. Obscure rarely used notions (e.g., the Medvedev lattice) and repetition of the same material in different contexts is avoided in favor of going through more complex constructions and results in the r.e.-degrees.