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For a limit cardinal $\alpha$ can we find a sequence of sets $(X_n)$ with $card X_1< card X_2<...< card X$and $card X= card X_1+ card X_2+...?$

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    $\begingroup$ What do the $\dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?) $\endgroup$ – Asaf Karagila Jan 3 '19 at 8:17
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Your question suggests you are looking for a countable sequence; then the answer is: NO. $\aleph_{\omega_1}$ is a limit cardinal, but not the sum of countably many smaller cardinals. You may want to study the notion `cofinality of a cardinal number'.

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