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 ${\mathrm{\aleph }}_0$

Work in ZF; no choice principle is used. There is a tower of ordinals with

$T(0)={\mathrm{\aleph }}_0,T(n+1)={\mathrm{\aleph }}_{T(n)}(n\in \omega ),$

so that $T(1)={\mathrm{\aleph }}_{{\mathrm{\aleph }}_0}={\mathrm{\aleph }}_{\omega }$ and $T(2)={\mathrm{\aleph }}_{{\mathrm{\aleph }}_{\omega }}$ (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 ${\mathrm{\aleph }}_1={\omega }_1$). Put

$\alpha =\sup \{T(n):n\in \omega \}=\bigcup \{T(n):n\in \omega \}\mathrm{.}$

Then

${\mathrm{\aleph }}_{\alpha }=\alpha ,\mathrm{cf}(\alpha )={\mathrm{\aleph }}_0,$

so $\alpha$ is an infinite cardinal (Cardinal (initial ordinal) and cardinality) fixed by the aleph operation, and it is singular (Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals).

So the aleph operation has fixed points, even though it is strictly increasing and satisfies $\beta \le {\mathrm{\aleph }}_{\beta }$ at every ordinal (The clauses at $0$, at a successor and at a limit determine exactly one operation $\alpha \mapsto {\mathrm{\aleph }}_{\alpha }$, in ZF, and — assuming the Axiom of Choice — exactly one operation $\alpha \mapsto {\mathrm{\beth }}_{\alpha }$; each value is an infinite cardinal, each is strictly increasing and continuous at limits, and $\alpha \le {\mathrm{\aleph }}_{\alpha }$). The power operation behaves differently: assuming the Axiom of Choice, so that $2^{\kappa }$ is a cardinal at all, $\kappa <2^{\kappa }$ at every cardinal (Assuming the Axiom of Choice, $2^{\kappa }=\mid \mathcal{P}(\kappa )\mid$, and Cantor's theorem in cardinal form: $\kappa <2^{\kappa }$), so there are no fixed points there at all. That comparison is an aside; nothing below uses it, and the example itself stays in ZF.

Why a fixed point is not a contradiction. $\beta \le {\mathrm{\aleph }}_{\beta }$ holds everywhere and the operation is strictly increasing, but neither forces $\beta <{\mathrm{\aleph }}_{\beta }$: strict increase compares the values at two different indices and says nothing about the value at one index. The tower converges exactly because at a limit the value is the supremum of the earlier ones, and the supremum of the tower is what the tower was climbing towards.

Nothing is chosen, and nothing beyond Replacement is used. The tower is a definable $\omega$-indexed family, Replacement makes its range a set, and every step of the verification is a computation. So the whole example is a theorem of ZF, like the aleph hierarchy it is built from.

Its cofinality is the smallest an infinite cardinal can have. $\mathrm{cf}(\alpha )={\mathrm{\aleph }}_0$ says $\alpha$ is reached by an $\omega$-indexed family, which is exactly how it was built. So this fixed point is singular, and, assuming the Axiom of Choice, by Assuming the Axiom of Choice: $\kappa <{\kappa }^{\mathrm{cf}(\kappa )}$ for every infinite cardinal $\kappa$, and $\mathrm{cf}(2^{\kappa })>\kappa$; in particular $\mathrm{cf}(2^{{\mathrm{\aleph }}_0})>{\mathrm{\aleph }}_0$ it is therefore not a possible value of $2^{{\mathrm{\aleph }}_0}$. Nothing above claims it is the least fixed point.