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| The \emph{beth numbers} are infinite cardinal numbers |
The \emph{beth numbers} are infinite cardinal numbers |
| defined in a similar manner to the aleph numbers, as described below. |
defined in a similar manner to the aleph numbers, as described below. |
| They are written $\beth_\alpha$, where $\beth$ is beth, |
They are written $\beth_\alpha$, where $\beth$ is beth, |
| the second letter of the Hebrew alphabet, |
the second letter of the Hebrew alphabet, |
| and $\alpha$ is an ordinal number. |
and $\alpha$ is an ordinal number. |
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| We define $\beth_0$ to be the first infinite cardinal (that is, $\aleph_0$). |
We define $\beth_0$ to be the first infinite cardinal (that is, $\aleph_0$). |
| For each ordinal $\alpha$, |
For each ordinal $\alpha$, |
| we define $\beth_{\alpha+1}=2^{\beth_\alpha}$. |
we define $\beth_{\alpha+1}=2^{\beth_\alpha}$. |
| For each limit ordinal $\delta$, |
For each limit ordinal $\delta$, |
| we define $\beth_\delta=\bigcup_{\alpha\in\delta}\beth_\alpha$. |
we define $\beth_\delta=\bigcup_{\alpha\in\delta}\beth_\alpha$. |
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| Note that $\beth_1$ is the cardinality of the continuum. |
Note that $\beth_1$ is the cardinality of the continuum. |
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| For any ordinal $\alpha$ the inequality $\aleph_\alpha\leq\beth_\alpha$ holds. |
For any ordinal $\alpha$ the inequality $\aleph_\alpha\leq\beth_\alpha$ holds. |
| The Generalized Continuum Hypothesis is equivalent to the assertion that |
The Generalized Continuum Hypothesis is equivalent to the assertion that |
| $\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$. |
$\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$. |
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| For every limit ordinal $\delta$, |
For every limit ordinal $\delta$, |
| the cardinal $\beth_\delta$ is a strong limit cardinal. |
the cardinal $\beth_\delta$ is a strong limit cardinal. |
| Every uncountable strong limit cardinal arises in this way. |
Every uncountable strong limit cardinal arises in this way. |
