# beth numbers

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

We define $\beth_{0}$ to be the first infinite cardinal (that is, $\aleph_{0}$). For each ordinal $\alpha$, we define $\beth_{\alpha+1}=2^{\beth_{\alpha}}$. For each limit ordinal $\delta$, we define $\beth_{\delta}=\bigcup_{\alpha\in\delta}\beth_{\alpha}$.

Note that $\beth_{1}$ is the cardinality of the continuum.

For any ordinal $\alpha$ the inequality $\aleph_{\alpha}\leqslant\beth_{\alpha}$ holds. The Generalized Continuum Hypothesis is equivalent to the assertion that $\aleph_{\alpha}=\beth_{\alpha}$ for every ordinal $\alpha$.

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

Title beth numbers BethNumbers 2013-03-22 14:17:09 2013-03-22 14:17:09 yark (2760) yark (2760) 9 yark (2760) Definition msc 03E10 AlephNumbers GeneralizedContinuumHypothesis