Assuming the Axiom of Choice: ${\mathrm{\beth }}_0={\mathrm{\aleph }}_0$, ${\mathrm{\beth }}_1=2^{{\mathrm{\aleph }}_0}=\mid \mathbb{R}\mid$, ${\mathrm{\beth }}_2=\mid \mathcal{P}(\mathbb{R})\mid$, and ${\mathrm{\beth }}_{\omega }$ has cofinality ${\mathrm{\aleph }}_0$

Assume the Axiom of Choice (The Axiom of Choice), which is what makes the beths available at all (The successor cardinal ${\kappa }^{+}$, the alephs ${\mathrm{\aleph }}_{\alpha }$, the beths ${\mathrm{\beth }}_{\alpha }$, successor and limit cardinals, and the identifications ${\mathrm{\aleph }}_0=\omega$ and ${\mathrm{\aleph }}_1={\omega }_1$). Then

${\mathrm{\beth }}_0={\mathrm{\aleph }}_0,{\mathrm{\beth }}_1=2^{{\mathrm{\aleph }}_0}=\mid \mathbb{R}\mid ,{\mathrm{\beth }}_2=2^{2^{{\mathrm{\aleph }}_0}}=\mid \mathcal{P}(\mathbb{R})\mid ,$

and

$\mathrm{cf}({\mathrm{\beth }}_{\omega })={\mathrm{\aleph }}_0<{\mathrm{\beth }}_{\omega },$

so ${\mathrm{\beth }}_{\omega }$ is singular (Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals).

The first three values are the definition unwound, once $\mid \mathbb{R}\mid =2^{{\mathrm{\aleph }}_0}$ is known ($\mathbb{R}\approx \mathcal{P}(\mathbb{N})$ in ZF, by the Cantor set for one injection and by the cuts $\{q\in \mathbb{Q}:q<x\}$ for the other; so $\mid \mathbb{R}\mid =2^{{\mathrm{\aleph }}_0}$ under the Axiom of Choice) and once $2^{\mid A\mid }=\mid \mathcal{P}(A)\mid$ is available for an arbitrary set (Assuming the Axiom of Choice, $2^{\kappa }=\mid \mathcal{P}(\kappa )\mid$, and Cantor's theorem in cardinal form: $\kappa <2^{\kappa }$). The last is the same cofinality argument that applies to ${\mathrm{\aleph }}_{\omega }$, and for the same reason: the index $\omega$ is a limit reached by an $\omega$-indexed family, and the beth operation is continuous at limits.

Where the beths and the alephs are known to agree, and where they are not. ${\mathrm{\beth }}_0={\mathrm{\aleph }}_0$ by the base clauses. Beyond that, this development proves ${\mathrm{\aleph }}_1\le {\mathrm{\beth }}_1$ (${\mathrm{\aleph }}_1\le 2^{{\mathrm{\aleph }}_0}$ under the Axiom of Choice, because $2^{{\mathrm{\aleph }}_0}$ is a cardinal strictly above ${\mathrm{\aleph }}_0$ and ${\mathrm{\aleph }}_1$ is the least such; so ${\omega }_1$ injects into $\mathbb{R}$) and nothing sharper: whether ${\mathrm{\beth }}_1={\mathrm{\aleph }}_1$ is the continuum hypothesis, which What each result on this page costs in choice, and where the continuum escapes what ZFC can decide records as undecided by the axioms in use here.

Why ${\mathrm{\beth }}_2$ is written two ways. As $2^{2^{{\mathrm{\aleph }}_0}}$ it is a cardinal computation; as $\mid \mathcal{P}(\mathbb{R})\mid$ it is a statement about a familiar set. The bridge is the general clause $2^{\mid A\mid }=\mid \mathcal{P}(A)\mid$ of Assuming the Axiom of Choice, $2^{\kappa }=\mid \mathcal{P}(\kappa )\mid$, and Cantor's theorem in cardinal form: $\kappa <2^{\kappa }$, applied at $A=\mathbb{R}$; without it the two expressions would have to be related by hand.

Singularity at a limit index is generic, not special to ${\mathrm{\beth }}_{\omega }$. The computation of step 2.2 uses only that the operation is strictly increasing and continuous at $\omega$, so it applies verbatim to ${\mathrm{\aleph }}_{\omega }$ ($\mathrm{cf}({\mathrm{\aleph }}_{\omega })={\mathrm{\aleph }}_0$, computed from the cofinal map $n\mapsto {\mathrm{\aleph }}_n$) and to the fixed-point tower of An ordinal $\alpha$ with ${\mathrm{\aleph }}_{\alpha }=\alpha$, built as the supremum of the tower ${\mathrm{\aleph }}_0,{\mathrm{\aleph }}_{{\mathrm{\aleph }}_0},{\mathrm{\aleph }}_{{\mathrm{\aleph }}_{{\mathrm{\aleph }}_0}},\dots$, and its cofinality is ${\mathrm{\aleph }}_0$. What is not generic is the value of the cofinality at other limit indices, which depends on the index and not on the operation.