The beth numbers, $\beth_\alpha$

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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 $|V_{\omega+\alpha}|=\beth_\alpha$, where $V_{\omega+\alpha}$ denotes the $\omega+\alpha$th stage of the cumulative hierarchy.

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. The continuum hypothesis is equivalent to the assertion that $\aleph_1=\beth_1$. The generalized continuum hypothesis 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 aleph fixed points, 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 $\forall\alpha >0(\beth_\alpha>\aleph_\alpha)$.