The independence of the continuum hypohtesis I just finished a short course on the basics of set theory (axioms, ordinals, cardinals, construction of real numbers, etc) but we did not talk anything about models.
I am interested in particular in the proof of the independence of the continuum hypothesis from ZFC and I have been recommended the book Set theory by Kenneth Kunen, the 2013 edition. Problem is this book seem to contain a lot more than what I need, just the background section has over 100 pages. Is there any "path" through the book that leads to this particular proof?
P.D. Any other source is  welcomed too. I just happen to have access to a printed copy of this book so it's easier.
Edit: I found this book Set theory and the continuum hypothesis by Cohen, it seems to fit perfectly well with my needs, it even has an introduction to logic as well as set theory and its only 150 pages, any opinions?
 A: I disagree with Andres Caicedo slightly, based on my own experience learning forcing: I think that the beginning of chapter 2 would be good to read. Learning how to use Martin's axiom will secretly teach you a lot about how forcing works, and in my experience is easier than diving straight into the proof.
Now, showing that CH is independent of ZFC consists of two pieces: showing that it's consistently true, and that it's consistently false. The only way to do the latter is via forcing. The former, however, has two methods: you can prove it with forcing, or by using $L$ (chapter VI, specifically section 4).
My suggestion is to do forcing for each part - that cuts down on the amount of reading you have to do, and also introduces you to two very different basic forcing arguments, based on two different important combinatorial properties: the countable chain condition, and countable closure.
As to forcing itself, after the first bit of chapter II you'll be ready to leap to chapter VII. Here Kunen's treatment is a bit opaque: I tried to self-study forcing, and wasn't able to (although once I got an intuition for the basic details Kunen was wonderful - you just need that first "push" to get started). I think here it might be good to take a break from Kunen and look at another source, perhaps the essay A beginner's guide to forcing, which takes a more informal tack; once you know what a forcing notion and a name are supposed to be doing, go back to Kunen, and you should be ready to tackle chapter $7$!
