In asking a question on this site about the unprovability on the Continuum Hypothesis, many people explained to me that for a given set of axioms, there are many different models that satisfy that set of axioms, a concept that I am very comfortable with now.
Hence, there exists a model for ZFC set theory that satisfies CH, and one that doesn't. (Correct me if I'm wrong about anything)
However, for practical purposes, it seems like there is model of set theory that is used almost universally, at least in branches of mathematics other than logic. It also seems like this model is "maximal" in the sense that if the existence of a set $A$ doesn't violate any of the axioms of ZFC, then $A$ exists in the universe of sets.
First off, am I even remotely correct in thinking this? Is there such a model? And if so, is it maximal in any sense?
If I am correct, then what's the big deal about the unprovability of the Continuum Hypothesis? It seems true of a lot of things. From reading the axioms, I don't see any reason why a set with the cardinality of the real numbers has to exist. And why be so squirrelly when talking about sets with cardinality between that of the reals and the natural numbers? If they exist within our maximal universe of sets, why not use them?
If I'm not correct, then does a maximal universe even exist? Or, is there a universe of sets, $U$, that satisfies ZFC, and all other models of ZFC are subsets of $U$?
If so, why wouldn't we use it? And my earlier questions still apply.
If not, why not? Is there a proof? And what model do we actually use, and why?
Sorry if this isn't a very well-asked question. It's kind of rambling and contains a lot of sub-questions, but I'm not sure how else to phrase this while still asking what I want to.