# Continuum

From Cantor's Attic

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 $\alpha$.