The relation between the limit cardinal $\alpha$ and a sequence of cardinal numbers strictly less than $\alpha$

Question

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

Answer 1

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'.