$\underline{\text{Some Background}}$: A classic axiomatic system of set theory in modernity is $ZFC$. Results from modern mathematical logic guarantee the existence of statements undecidable given sufficiently complex axiom schemas like $ZFC$, such as $CH$, the continuum hypothesis. Some people then choose to study $ZFC + CH$, since it is generally perceived as more intuitive that there should be no set which is too big to be countable but is strictly smaller than $|\mathbb{R}|$.
$\underline{\text{The Question}}$: However, this does not mean there there isn't a perfectly nice model and associated set theory which has these strange sets. My question is this; what information exists about set theories with axioms $ZFC + (\neg CH)$? Does anyone know anything about this set theory, about results and theorems from this type of set theory, are people studying it, might there even be some sort of strange text book written which contains a large digression to set theories of this sort?
To me this would be something very interesting to read about, and I decided to post this question in hopes that I'm not the only one, and so that perhaps others in the future have an easier time finding a starting point for learning about these set theories if one does indeed exist.