I will present some results concerning the following general problems:
Let $\mathcal{K}$ be a class of compact spaces.
1. How to classify (up to isomorphisms) Banach spaces $C(K)$ of real-valued continuous functions on $K$ (with supremum norm), for $K\in \mathcal{K}$?
2. How many isomorphic types of $C(K)$ are there, for $K\in \mathcal{K}$?
The classical result of Bessaga and Pełczyński gives us a complete classification of $C(K)$, for the class of countable compact spaces $K$; in particular, we have $\omega_1$ isomorphic types of such spaces $C(K)$. On the other hand, Milutin's theorem says that, for the class of uncountable metrizable compact spaces $K$, we have only one isomorphic type of spaces $C(K)$.
I will discuss two well-known classes of compact spaces of weight $\omega_1$, for which problem 2 is not decidable in ZFC.
The first of these classes is the class $\mathcal{K}$ of compact spaces generated by families of almost disjoint subsets of the set of natural numbers $\omega$, usually associated with the names of Mrówka, Isbell, Franklin, or Aleksandrov and Urysohn.
Assuming the continuum hypothesis, we have $2^{2^\omega}$ isomorphic types of $C(K)$, for $K\in \mathcal{K}$. In turn, assuming Martin's axiom and negation of the continuum hypothesis, for all $K,L \in \mathcal{K}$ with $w(K) = w(L) = \omega_1$, the spaces $C(K)$ and $C(L)$ are isomorphic (joint results with R. Pol, F. Cabello S\'anchez, J. Castillo, G. Plebanek, A. Salguero-Alarc\'on).
The second class considered is the class $\mathcal{L}$ of separable, compact linearly ordered spaces of weight $\omega_1$.
Again, assuming the continuum hypothesis, we have $2^{2^\omega}$ isomorphic types of $C(K)$, for $K\in \mathcal{L}$. On the other hand, assuming a certain axiom proposed by Baumgartner, we have only one class of isomorphic types $C(K)$, for $K\in \mathcal{L}$ (joint results with M. Korpalski).