Beths

Definition. Beth numbers are defined by transfinite recursion:

1. \(\beth_0 = \aleph_0\),
2. \(\beth_{\alpha+1} = 2^{\beta_\alpha}\) where \(\alpha\) is an ordinal,
3. \(\beth_\lambda = \sup\{\beth_\alpha:\alpha < \lambda \}\) where \(\lambda\) is a limit ordinal.

Thus Beths are the cardinalities of sets obtained by recursively taking the power set operation (when the associated ordinal is not a limit ordinal). The second beth number \(\beth_1\) is equal to the cardinality of the continuum.

Assuming the axiom of choice, cardinals are linearly ordered, and every two cardinalities are comparable. By definition no infinite cardinalities are between \(\aleph_0\) and \(\aleph_1\) it follows that \(\beth_1 \geq \aleph_1\) and also \(\beth_\alpha \geq \aleph_\alpha\) for all \(\alpha\) ordinals.

The continuum hypothesis is equivalent to the claim

\[\beth_1 = \aleph_1.\]

The generalized continuum hypothesis is the claim that for all ordinals \(\alpha\), \(\beta_\alpha = \aleph_\alpha\).