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[parent] beth numbers (Definition)

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.



"beth numbers" is owned by yark.
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See Also: aleph numbers, generalized continuum hypothesis


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Cross-references: uncountable, strong limit cardinal, generalized continuum hypothesis, inequality, cardinality of the continuum, limit ordinal, ordinal number, aleph numbers, cardinal numbers, infinite
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This is version 6 of beth numbers, born on 2004-04-02, modified 2006-12-30.
Object id is 5740, canonical name is BethNumbers.
Accessed 1913 times total.

Classification:
AMS MSC03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

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