In https://en.wikipedia.org/wiki/Buchholz_psi_functions it is written : "If $\alpha$ is a limit ordinal then $cof(\alpha) \in \lbrace \omega \rbrace \cup \lbrace \Omega_{\mu+1} | \mu \ge 0 \rbrace$".
Does it mean that a limit ordinal cannot have a cofinality of $\Omega_\omega$, and why ?
Because the cofinality of $\aleph_{\omega}$ itself is $\omega$, so if there is an unbounded subset $A \subseteq \alpha$ of size $\aleph_{\omega}$, then there is also an unbounded subset of size $\omega$.