There is a very curious theorem by Shelah which states that:
If for every natural number $n$ we have $2^{\aleph_n} < \aleph_{\omega}$ then, $2^{\aleph_{\omega}} < \aleph_{\omega_4}$.
An other way to state this is this one: if $\aleph_{\omega}$ is a strong limit cardinal then: $$\prod_{n=0}^{\infty} \aleph_n < \aleph_{\omega_4} $$