I am looking to start seriously self-studying mathematical logic, and, first, I am wondering if any of you can recommend a sequence of say 3-4 textbooks on logic, where the first book would focus heavily on intuition, the second book on the basics, the third book on "advanced" material (think graduate texts in math), and the fourth book(s) on modern logic research (I am currently most interested in computability theory)?
To this end, as I have searched for good books to read, I have noticed that most of the logic textbooks were published decades ago. For example, I have seen Monk's Mathematical Logic get recommended a number of times, but it was published in the 70's, so I am unclear on how relevant the material is today, and whether it would be worth my time (as a Ph.D. student). Generally speaking, should I be focusing on reading books published in say the last three decades, or are older books still relevant and worth reading? For example would reading Church's Introduction to Mathematical Logic, or Kleene's Introduction to Meta-Mathematics be worth it?
In particular, would reading old logic textbooks just introduce you to areas and concepts that are not relevant for today's research?
