In the case of exponentiation of singular cardinals, ZFC has a lot more to say. In 1989, Shelah proved the remarkable result that if \(\aleph_\omega\) is a strong limit, that is, \(2^{\aleph_n}&lt;\aleph_\omega\), for every \(n&lt;\omega\), then \(2^{\aleph_\omega}&lt;\aleph_{\omega_4}\) (see Shelah (1994)). The technique developed by Shelah to prove this and similar theorems, in ZFC, is called pcf theory (for possible cofinalities), and has found many applications in other areas of mathematics.