= Beth omega is a singular strong limit above the continuum
{c}
{title2=$\operatorname{cf}(\beth_\omega)=\omega,\quad 2^\omega<\beth_\omega$}
The <Beth numbers> $\beth_0=\aleph_0$, $\beth_{n+1}=2^{\beth_n}$ form a strictly increasing countable sequence by <Cantor's theorem>. Their supremum $\beth_\omega$ has <cofinality> $\omega$ because no finite subset is cofinal. It exceeds $2^{\aleph_0}=\beth_1$. It is also a <strong limit cardinal>: if $\rho<\beth_\omega$, some $n$ has $\rho<\beth_n$, so $2^\rho\leq\beth_{n+1}<\beth_\omega$. Thus the <singular cardinal> example needs no <Continuum hypothesis>.
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