= Solution
For $\kappa=\aleph_\omega$, monotonicity of <cardinal exponentiation> gives
$$
2^\kappa\leq\aleph_0^\kappa\leq(2^{\aleph_0})^\kappa
=2^{\aleph_0\cdot\kappa}=2^\kappa.
$$
The middle equality uses the <currying law for cardinal exponentiation>, and the final one uses $\aleph_0\cdot\kappa=\kappa$. Hence $\aleph_0^\kappa=2^\kappa$, even before assuming GCH. Applying the <Generalized continuum hypothesis> yields
$$
\boxed{\aleph_0^{\aleph_\omega}=\aleph_{\omega+1}.}
$$
Back to article page
