= GCH arithmetic at aleph omega
{c}
{title2=$\aleph_\omega^{\aleph_\omega}=\prod_{n<\omega}\aleph_n=\aleph_\omega^{\aleph_0}=\aleph_{\omega+1}$}
Put $\kappa=\aleph_\omega$. <ZFC> gives $\kappa^\kappa=2^\kappa$. König's theorem and the shift of product coordinates give $\kappa<\prod_n\aleph_n\le\kappa^{\aleph_0}\le2^\kappa$. The single instance $2^\kappa=\kappa^+$ of the <Generalized continuum hypothesis>[generalized continuum hypothesis] therefore fixes all three displayed values. The strict lower bound explains why <countable> exponentiation cannot leave this <singular cardinal> unchanged.
Back to article page
