# What is the sup of the cardinalities of the chains in $\mathcal P(X)$?

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

• – 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