We have the following implication:
Any countable union of countable sets is countable $\rightarrow$ $\omega_1$ is regular $\rightarrow$ $\mathcal{P}(\omega)$ is not a countable union of countable sets.
According to the Choiceless Grapher, these implications are not reversible. But I can't seem to locate the references. Can someone point me to where these non-reversibility results are located? Thank you!
On
The fact that $38$ does not imply $34$ can be found in Zermelo's Axiom of Choice by G.H. Moore ($1982$) page $324$. I found this by looking in the book CGraph is based off of. I couldn't find anything about whether or not $34$ implies $31$. It seems this could be an open problem. At the very least I don't think the book mentions it.
For the first thing, any symmetric extension built with either Cohen reals, or without adding new reals (or generally without collapsing $\omega_1$) will have $\omega_1$ regular. As such it's easy to arrange a model where the reverse implication of the first arrow fails. So, for example, the second Cohen model where there is a countable union of pairs without a choice function. You can also arrange this in the level of the reals by adding a countable sequence of countable sets of reals and not having a uniform enumeration of all these new sets.
The second one is trickier, but here are two ways:
The Truss' model is a model where $\omega_1$ is regular, but $\Bbb R$ is not the countable union of countable sets. The idea is to mimic the Solovay model, but to start with a singular cardinal and not necessarily an inaccessible cardinal.
Use the Feferman–Levy model where we collapse $\aleph_\omega$ and also add $\aleph_{\omega+1}$ Cohen reals (before or after, it doesn't matter since it's a well-ordered forcing in any model of $\sf ZF$ it will not destroy "families without a choice function"). Or generally first blow the continuum above $\aleph_\omega$ to have a regular cardinality (just to simplify things) and then collapse $\aleph_\omega$ to be $\omega_1$.