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What is the sup of the cardinalities of the chains in $\mathcal P(X)$, where $X$ is a set?

Here chain means totally ordered set, and $\mathcal P(X)$ is the power set of $X$ (ordered by inclusion).

We can assume that $X$ is infinite, because otherwise the answer is obvious.

We can assume that $X$ is uncountable, because otherwise the answer is given in these posts of Asaf Karagila and Noah Schweber.

Clearly, the sup $s$ in question satisfies $\operatorname{Card}(X)\le s\le2^{\operatorname{Card}(X)}$. (If $X$ is countable we have $s=2^{\operatorname{Card}(X)}$.)

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This post of Eric Wofsey answers the question.

I'm posting this answer as a community wiki, and I'm panning to accept it as soon as possible in order that the question be considered as answered. If you think the question should be closed as a duplicate, please let me know, and kindly tell me what I should do.

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