TITLE: Some "axiom of choice" and "dependent choice" issues
QUESTION [10 upvotes]: I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with ZF+DC, but other forms of choice that are not implied by DC can be more controversial.
[Therefore it seems natural that people should give higher priority to discussing the differences in provable theorems between ZFC and ZF+DC -- or at least, the differences in provable theorems between ZFC and ZF+(countable choice) -- than to discussing the differences in provable theorems between ZFC and ZF. (Indeed, you basically can't do any analysis in just ZF.)]
My questions are:

Is it consistent with ZF+DC that every subset of $\mathbb{R}$ is Borel-measurable?
If the answer to Q1 is no: Is it consistent with ZF+DC that a countably generated $\sigma$-algebra can have a cardinality strictly larger than that of the continuum?
Is it a theorem of ZF+DC that there exists an injective map from the set $\omega_1$ of well-orderings of $\mathbb{N}$ into $\mathbb{R}$?

Thanks.

REPLY [3 votes]: Let me add that the answer to (3) is negative, if you are willing to assume that inaccessible cardinals are consistent.
We say that $\omega_1$ is inaccessible to reals if for every real $x$, $\omega_1^{L[x]}<\omega_1$. This implies that $\omega_1$ is a limit cardinal in $L$. But if we also assume $\sf DC$, then $\omega_1$ is regular, in which case it is also regular in $L$. The conjunction of the two means that it is inaccessible there.
It is consistent that there is no injection from $\omega_1$ into the reals, e.g. in Solovay's model, where $\sf DC$ holds. And on the other hand, if $\omega_1$ is accessible to reals, then for some real number $x$, there is an injection from $\omega_1$ into the reals constructible from $x$, and in particular into $\Bbb R$.
So we see that it is consistent that $\omega_1$ is inaccessible to reals, and that there is no injection from $\omega_1$ into $\Bbb R$. (Of course it is consistent that $\omega_1$ is inaccessible to reals and there is such an injection. Simply look at the model obtained by collapsing an inaccessible to be $\omega_1$.)