Chain of length $2^{\aleph_0}$ in $ (P(\mathbb{N}),\subseteq)$

elementary-set-theory order-theory

Hint: Since there is a bijection between $\mathbb Q$ and $\mathbb N$ there is an order isomorphism between their power sets with inclusion.

Now think about Dedekind-cuts.

Also, your chain is indeed countable.

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