Another area in which large cardinals play an important role is the exponentiation of singular cardinals. The so-called Singular Cardinal Hypothesis (SCH) completely determines the behavior of the exponentiation for singular cardinals, modulo the exponentiation for regular cardinals. The SCH follows from the GCH, and so it holds in \(L\). A consequence of the SCH is that if \(2^{\aleph_n}&lt;\aleph_\omega\), for all finite \(n\), then \(2^{\aleph_{\omega}}=\aleph_{\omega +1}\). Thus, if the GCH holds for cardinals smaller than \(\aleph_\omega\), then it also holds at \(\aleph_\omega\). The SCH holds above the first supercompact cardinal (Solovay). But Magidor (1977) showed that, remarkably, assuming the existence of large cardinals it is possible to build a model of ZFC where the GCH first fails at \(\aleph_\omega\), hence the SCH fails. Large cardinals stronger than measurable are actually needed for this. In contrast, however, ZFC alone suffices to prove that if the SCH holds for all cardinals smaller than \(\aleph_{\omega_1}\), then it also holds for \(\aleph_{\omega_1}\). Moreover, if the SCH holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals (Silver).