GCH arithmetic at aleph omega (${\aleph }_{\omega }^{{\aleph }_{\omega }}={\prod }_{n<\omega }{\aleph }_n={\aleph }_{\omega }^{{\aleph }_0}={\aleph }_{\omega +1}$)

Put $\kappa ={\aleph }_{\omega }$. ZFC gives ${\kappa }^{\kappa }=2^{\kappa }$. König's theorem and the shift of product coordinates give $\kappa <{\prod }_n{\aleph }_n\le {\kappa }^{{\aleph }_0}\le 2^{\kappa }$. The single instance $2^{\kappa }={\kappa }^{+}$ of the generalized continuum hypothesis therefore fixes all three displayed values. The strict lower bound explains why countable exponentiation cannot leave this singular cardinal unchanged.