The *continuum* is the cardinality of the reals $\mathbb{R}$, and is variously denoted $\frak{c}$, $2^{\aleph_0}$, $\|\mathbb{R}\|$, $\beth_1$, $2^\omega$. This article is a stub. Please help us to improve Cantor's Attic by adding information. ## Continuum hypothesis The *continuum hypothesis* is the assertion that the continuum is the same as the first uncountable cardinal <a href="Aleph_one" class="mw-redirect" title="Aleph one">$\aleph_1lt;/a>. The *generalized continuum hypothesis* is the assertion that for any infinite cardinal $\kappa$, the power set $P(\kappa)$ has the same cardinality as the <a href="Successor" class="mw-redirect" title="Successor">successor</a> cardinal $\kappa^+$. This is equivalent, by transfinite induction, to the assertion that $\aleph_\alpha=\beth_\alpha$ for every ordinal $\alpha$.