Let $S \subset \mathcal P \Bbb N$ with the condition that if $A, B \in S$ then $A \subset B$ or $B \subset A$. Can $S$ be uncountable?
I've been thinking about this problem for a while and I think it's probably the case that $S$ can be uncountable, but I'm not sure how to prove it. If that's not the case, please correct me.
I would prefer hints to a whole solution please, because I'm really struggling to understand how one approaches this problem in the first place, and any suggestions about these thought-processes would be much appreciated. I've come across the useful heuristic of thinking of countable sets as those whose elements can be given specific finite descriptions, but otherwise I don't have much intuition here.
Thank you very much in advance.