beth numbers

The beth numbers are infinite cardinal numbers defined 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.

Title	beth numbers
Canonical name	BethNumbers
Date of creation	2013-03-22 14:17:09
Last modified on	2013-03-22 14:17:09
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Definition
Classification	msc 03E10
Related topic	AlephNumbers
Related topic	GeneralizedContinuumHypothesis