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

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

• What do the $\dots$ mean here? (Yes, I get it, an infinite sequence, but how long exactly?) – Asaf Karagila Jan 3 '19 at 8:17

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'.