Chain of Natural Number Sets

[A puzzle which requires knowing what countable/uncountable mean]

Let $\mathbb{N}$ be the set of naturals: $\{1,2,\dots\}$.

Suppose $F$ is a subset of the power-set of $\mathbb{N}$ (i.e. $F$ is a set of subsets of $\mathbb{N}$), such that for any two distinct sets $A, B \in F$, either $A \subset B$ or $B \subset A$.

Basically $F$ is a chain of sets, each containing the previous one.

Is there an $F$ which is uncountable?


[Solution]