Paul Cohen told us that whether or not there is $S$ with \begin{equation} \aleph_0<|S|<2^{\aleph_0} \end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two distinct mathematical worlds, one with such an $S$, the other without such sets.
What differences between theses two worlds have we discovered so far? Does this affect subjects other than logic?
Thanks!
There are many differences, and not just in set theory or logic. Several questions both here and on MO address this. Several examples are mentioned here, where we list a few statements (from algebra, order theory, and analysis) equivalent to $\mathsf{CH}$.
A key difference is in the behavior of uncountable ordered sets: As mentioned here, under $\mathsf{CH}$ there are $2^{|\mathbb R|}$ uncountable dense subset of $\mathbb R$ none of which embeds into any of the others. On the other hand, it is consistent with the failure of $\mathsf{CH}$ that there are precisely $5$ uncountable ordered sets, such that given any uncountable linear order $\ell$, at least one of these five orders embeds into $\ell$ and, in particular, any two uncountable dense subsets of $\mathbb R$ are isomorphic.
And this is typical: $\mathsf{CH}$ tends to give us a very "chaotic" universe (there are many uncountable ordered sets, there are discontinuous homomorphisms between Banach algebras, there are many outer automorphisms of Calkin algebras, etc). On the other hand, the negation of $\mathsf{CH}$ is consistent with very strong "classification" theorems that indicate that this chaotic behavior is impossible. A very good paper addressing this ("Combinatorial dichotomies in set theory"), by Todorcevic, can be found here. Now, granted, typically it is not the negation of $\mathsf{CH}$ alone that gives us these classification results. But, in the absence of $\mathsf{CH}$, mathematicians tend to adopt appropriate alternatives ("strong forcing axioms").
A remark probably worth mentioning is that there is a wealth of interesting mathematics that becomes "invisible" in the presence of strong arithmetic restrictions such as $\mathsf{GCH}$ or even just $\mathsf{CH}$. For example, many interesting results in cardinal arithmetic are trivial under $\mathsf{GCH}$. Also, the study of cardinal invariants of the continuum trivializes under $\mathsf{CH}$ (all uncountable invariants now being $|\mathbb R|$). (Here is a list of questions or answers on this site mentioning cardinal characteristics.)
