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The question is inspired by a relatively easy question in my problem sheet

Let $S$ be a set of subsets of $\mathbb N$ such that $\forall A,B\in S$, either $A\subset B$ or $B\subset A$. Must $S$ be countable?

The answer is of course no. Just replace $\mathbb N$ by $\mathbb Q$ and consider the family $S=\{A_x\}_{x\in\mathbb R}$ where $A_x=\{q\in\mathbb Q:q<x\}$. This problem aside, however, one can bring up a generalised question:

Let $X$ be a set. Must there be an inclusion chain $C\subset 2^X$ such that $|C|>|X|$? If so, must there be such a $C$ with $|C|=|2^X|$?

The approach to the motivation problem cannot be applied directly as it uses the established result of $\mathbb Q$ being dense in $\mathbb R$.

This makes the problem look significantly more difficult as one can even struggle in the case where $X=\mathbb R$. I have made some assorted attempts but still cannot find a complete proof.

Any help would be appreciated.

BAI
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    First part is a duplicate of https://math.stackexchange.com/questions/2078115/existence-of-large-chains-provable-in-zfc. Second part is a duplicate of https://mathoverflow.net/questions/48231/given-a-cardinal-k-whats-the-biggest-dense-linear-order-with-a-dense-subset-of. – Eric Wofsey Dec 15 '19 at 16:00
  • @EricWofsey Thank you very much for the reference! – BAI Dec 15 '19 at 16:12

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