Finding an uncountable chain of subsets the integers
Solution 1:
As mentioned in Uncountable chains (in the question itself), one can do this as follows:
Pick a bijection between $\mathbb{N}$ and $\mathbb{Q}$ so it suffices to find such a chain in the powerset of $\mathbb{Q}$. But $\mathbb{Q}\subseteq\mathbb{R}$ so for any real number $r$ we can define the subset $\Gamma_r = \{x\in\mathbb{Q}\mid x < r\}$. This then defines an injective order-preserving map from $\mathbb{R}$ to $\mathcal{P}(\mathbb{Q})$ (given by $r\mapsto \Gamma_r$), which gives us the uncountable chain.