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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 $s=2^{\operatorname{Card}(X)}$.)

Pierre-Yves Gaillard
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  • See https://math.stackexchange.com/questions/2078115/existence-of-large-chains-provable-in-zfc and https://mathoverflow.net/questions/48231/given-a-cardinal-k-whats-the-biggest-dense-linear-order-with-a-dense-subset-of. – Eric Wofsey Jun 19 '17 at 20:02
  • @EricWofsey - Thanks! I see that this is an old and classic problem. Perhaps the question should be closed as a duplicate. - It seems to me that this post of yours answers my question above (in the sense that it summarizes what's presently known about the problem). – Pierre-Yves Gaillard Jun 19 '17 at 20:44

1 Answers1

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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.

Pierre-Yves Gaillard
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