I have seen a proof that $|\mathcal{P}(\mathbb{N})| \neq \aleph_\omega$ using the fact that $\aleph_\omega$ is the union of countably many smaller cardinals, while $|\mathcal{P}(\mathbb{N})|$ is not. Is it consistent with ZFC that $|\mathcal{P}(\mathbb{N})| > \aleph_\omega$?