Continuum

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 $\alpha$.