I’ve just finished reading Timothy Chow’s A beginner’s guide to forcing and what a nice read! I recommend all readers, particularly those interested in model and set theory, to take a look.
I’m particularly fond of the exposition at the beginning addressing the validity of ZFC as the foundation of maths, which I had had qualms about at some point:
One common confusion about models of ZFC stems from a tacit expectation that some people have, namely that we are supposed to suspend all our preconceptions about sets when beginning the study of ZFC. For example, it may have been a surprise to some readers to see that a universe is defined to be a set together with. . . . Wait a minute—what is a set? Isn’t it circular to define sets in terms of sets? In fact, we are not defining sets in terms of sets, but universes in terms of sets. Once we see that all we are doing is studying a subject called “universe theory” (rather than “set theory”), the apparent circularity disappears.
The reader may still be bothered by the lingering feeling that the point of introducing ZFC is to “make set theory rigorous” or to examine the foundations of mathematics. While it is true that ZFC can be used as a tool for such philosophical investigations, we do not do so in this paper. Instead, we take for granted that ordinary mathematical reasoning—including reasoning about sets—is perfectly valid and does not suddenly become invalid when the object of study is ZFC. That is, we approach the study of ZFC and its models in the same way that one approaches the study of any other mathematical subject.
Also there is a very nice, if not the best I’ve seen, explanation of Skolem’s paradox. So much so for the introduction part, I’ll leave you to find out what forcing is and how it is used to prove the independence of CH from ZFC.