A while ago, I took a long break from set theory and came back recently. Now, I've been more able to understand concepts which used to be completely foreign to me, for example ultrapowers and mice in inner model theory.
I'm still having trouble understanding the "intuition" of forcing, and how one is supposed to come up with models by forcing. Even though I understand each individual concept, and I can read about the proofs, I can't really understand the whole concept as a unit.
I'm not really looking for a guide as a "beginner's forcing book," more just an explanation of how one might create a forcing notion to solve a certain problem. For example, how did Cohen know which poset to force with to create a model in which CH is false?
Sorry if this question sounds a bit ambiguous, I'm not the best at putting things into words. The point is, I don't quite understand how to come up with specific posets for certain problems.