In ZFC we know that the continuum cannot have cofinality $\omega$. 


However, in the Feferman-Levy model we have that $\frak c=\aleph\_1$, and that $\operatorname{cf}(\omega\_1)=\omega$. In fact in the Feferman-Levy model, $\aleph\_\omega^L=\aleph\_1^V$. 


Is it consistent with ZF that $\frak c=\aleph\_\omega$? Does that mean that the only restriction in ZF on the cardinality of the continuum is $\aleph\_0<\frak c$?



Is it consistent relative to ZF that $\frak c = \aleph_\omega$?
The answer is no. The continuum cannot be $\aleph\_\omega$, and this can be proved in ZF, that is, without using the axiom of choice. To see
this, suppose towards contradiction that $P(\omega)$ is equinumerous with
$\aleph\_\omega$. Since $P(\omega)$ is equinumerous
with $P(\omega)^\omega$, and this does not require AC, it follows
that there is a bijection $f:\aleph\_\omega\cong
(\aleph\_\omega)^\omega$. Let $g(n)$ be the first ordinal
$\alpha\lt\aleph\_\omega$ that is not among $f(\beta)(n)$
for any $\beta\lt\aleph\_n$. Since there are fewer than
$\aleph\_\omega$ many such $\beta$, it follows that there
are fewer than $\aleph\_\omega$ many such $f(\beta)(n)$, and
so such an $\alpha$ exists. Thus, $g:\omega\to
\aleph\_\omega$. But notice that for any particular
$\alpha\lt\aleph\_\omega$, we have $\alpha\lt\aleph\_n$ for
some $n$ and consequently $g(n)\neq f(\alpha)(n)$, and thus
$g\neq f(\alpha)$. Thus, $f$ was not surjective to
$(\aleph\_\omega)^\omega$ after all, a contradiction.


This is just a standard proof of Konig's theorem (that
$\aleph\_\omega^\omega\gt\aleph\_\omega$), and the point is
that it doesn't use AC.


