# Difference between revisions of "Continuum"

From Cantor's Attic

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The ''continuum'' is the cardinality of the reals $\mathbb{R}$, and is variously denoted $\frak{c}$, $2^{\aleph_0}$, $|\mathbb{R}|$, $\beth_1$, $2^\omega$. | The ''continuum'' is the cardinality of the reals $\mathbb{R}$, and is variously denoted $\frak{c}$, $2^{\aleph_0}$, $|\mathbb{R}|$, $\beth_1$, $2^\omega$. | ||

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== Continuum hypothesis == | == Continuum hypothesis == | ||

The ''continuum hypothesis'' is the assertion that the continuum is the same as the first uncountable cardinal [[aleph one | $\aleph_1$]]. The ''generalized continuum hypothesis'' is the assertion that for any infinite cardinal $\kappa$, the power set $P(\kappa)$ has the same cardinality as the [[successor]] cardinal $\kappa^+$. This is equivalent, by transfinite induction, to the assertion that $\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$. | The ''continuum hypothesis'' is the assertion that the continuum is the same as the first uncountable cardinal [[aleph one | $\aleph_1$]]. The ''generalized continuum hypothesis'' is the assertion that for any infinite cardinal $\kappa$, the power set $P(\kappa)$ has the same cardinality as the [[successor]] cardinal $\kappa^+$. This is equivalent, by transfinite induction, to the assertion that $\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$. |

## Revision as of 04:46, 3 January 2012

The *continuum* is the cardinality of the reals $\mathbb{R}$, and is variously denoted $\frak{c}$, $2^{\aleph_0}$, $|\mathbb{R}|$, $\beth_1$, $2^\omega$.

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## Continuum hypothesis

The *continuum hypothesis* is the assertion that the continuum is the same as the first uncountable cardinal $\aleph_1$. The *generalized continuum hypothesis* is the assertion that for any infinite cardinal $\kappa$, the power set $P(\kappa)$ has the same cardinality as the successor cardinal $\kappa^+$. This is equivalent, by transfinite induction, to the assertion that $\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$.