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Given a set $X$, let $\text{len}(X)$ denote the maximal cardinality of a chain in $\mathcal{P}(X)$. For example, for finite $n$, it's clear that $\text{len}(\{1, \ldots, n\}) = n + 1$. On the other hand, we have $\text{len}(\mathbb{N}) = |\mathbb{R}|$. Is it generally true that for infinite $X$, $\text{len}(X) = |\mathcal{P}(X)|$? Or does there exist some infinite $X$ with $\text{len}(X) < |\mathcal{P}(X)|$?

Smiley1000
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    Unfortunately, the answers in the above link only give affirmative answers under additional set theoretic assumptions, as (GCH). This MO post might provide more complete answers, in particular, it also gives negative answers, if we should believe the comments (of course, only consistently, as well). – Ulli Oct 27 '24 at 09:07

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