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This question already has an answer here:

Does there exists an uncountable infinite chain in P(ℕ) (subsets of natural numbers) ordered by inclusion?

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marked as duplicate by Ross Millikan, Asaf Karagila elementary-set-theory Jan 24 at 19:54

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Indeed. As an example, consider the images of Dedekind cuts via the bijection between $\mathbb{Q}$ and $\Bbb{N}$.

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