The *beth numbers* $\beth_\alpha$ are defined by transfinite recursion: - $\beth_0=\aleph_0$ - $\beth_{\alpha+1}=2^{\beth_\alpha}$ - $\beth_\lambda=\sup_{\alpha\lt\lambda}\beth_\alpha$, for limit ordinals $\lambda$ Thus, the beth numbers are the cardinalities arising from iterating the power set operation. It follows by a simple recursive argument that $\|V_{\omega+\alpha}\|=\beth_\alpha$. ## Beth one The number $\beth_1$ is $2^{\aleph_0}$, the cardinality of the power set $P(\aleph_0)$, which is the same as the [continuum](Continuum.md "Continuum"). The <a href="Continuum_hypothesis" class="mw-redirect" title="Continuum hypothesis">continuum hypothesis</a> is equivalent to the assertion that $\aleph_1=\beth_1$. The <a href="GCH" class="mw-redirect" title="GCH">generalized continuum hypothesis</a> is equivalent to the assertion that $\beth_\alpha=\aleph_\alpha$ for all ordinals $\alpha$. ## Beth omega The cardinal $\beth_\omega$ is the smallest uncountable cardinal exhibiting the interesting property that whenever a set $X$ has cardinality less than $\beth_\omega$, then also the power set $P(X)$ also has size less than $\beth_\omega$. ## Strong limit cardinal More generally, a cardinal $\kappa$ is a *strong limit cardinal* if whenever $\gamma\lt\kappa$, then $2^\gamma\lt\kappa$. Thus, the strong limit cardinals are those cardinals closed under the exponential operation. The strong limit cardinals are precisely the cardinals of the form $\beth_\lambda$ for a limit ordinal $\lambda$. ## Beth fixed point A cardinal $\kappa$ is a *$\beth$-fixed point* when $\kappa=\beth_\kappa$. Just as in the construction of <a href="Aleph_fixed_point" class="mw-redirect" title="Aleph fixed point">aleph fixed points</a>, we may similar construct beth fixed points: begin with any cardinal $\beta_0$ and let $\beta_{n+1}=\beth_{\beta_n}$; it follows that $\kappa=\sup_n\beta_n$ is a $\beth$-fixed point, since $\beth_\kappa=\sup_n\beth_{\beta_n}=\sup_n\beta_{n+1}=\kappa$. One may similarly construct $\beth$-fixed points of any desired cardinality, and indeed, the class of $\beth$-fixed points are precisely the closure points of the function $\alpha\mapsto\beth_\alpha$ and therefore form a closed unbounded proper class of cardinals. Every $\beth$-fixed point is an $\aleph$-fixed point as well. Since every model of ZFC satisfies the existence of a $\beth$-fixed point, it follows that no model of ZFC satisfies $\forall\alpha >0(\beth_\alpha>\aleph_\alpha)$.