beth numbers

The beth numbers are infinite cardinal numbersdefined in a similar manner to the aleph numbers, as described below.They are written ${\mathrm{\beth }}_{\alpha }$, where $\mathrm{\beth }$ is beth,the second letter of the Hebrew alphabet,and $\alpha$ is an ordinal number.

We define ${\mathrm{\beth }}_0$ to be the first infinite cardinal (that is, ${\mathrm{\aleph }}_0$).For each ordinal $\alpha$,we define ${\mathrm{\beth }}_{\alpha +1}=2^{{\mathrm{\beth }}_{\alpha }}$.For each limit ordinal $\delta$,we define ${\mathrm{\beth }}_{\delta }={\bigcup }_{\alpha \in \delta }{\mathrm{\beth }}_{\alpha }$.

Note that ${\mathrm{\beth }}_1$ is the cardinality of the continuum.

For any ordinal $\alpha$ the inequality ${\mathrm{\aleph }}_{\alpha }⩽{\mathrm{\beth }}_{\alpha }$ holds.The Generalized Continuum Hypothesis is equivalent to the assertion that${\mathrm{\aleph }}_{\alpha }={\mathrm{\beth }}_{\alpha }$ for every ordinal $\alpha$.

For every limit ordinal $\delta$,the cardinal ${\mathrm{\beth }}_{\delta }$ is a strong limit cardinal.Every uncountable strong limit cardinal arises in this way.