Some "axiom of choice" and "dependent choice" issues

Question

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:

1. Is it consistent with ZF+DC that every subset of $\mathbb{R}$ is Borel-measurable?
2. 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?
3. 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.

Answer 1

To elaborate a bit on Ashutosh's comment, my answer to this question shows that even countable AC, which is strictly weaker than DC, suffices to show that most sets of reals are not Borel. Thus, the answer to question (1) is negative. The argument there shows also that the answer to question (2) is negative, since one can code the sets by well-founded trees, just as Ashutosh mentions in his comments.

Answer 2

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 $\omega_1$.)

Answer 3

The answer to (2) actually depends on the exact meaning of "strictly larger", since cardinalities without choice are a mess. Working in ZF+DC, if $\mathcal{B}$ is a countably generated $\sigma$-algebra, there is always a surjection from $\mathcal{P}(\omega)$ to $\mathcal{B}$, but it could happen that $|\mathcal{B}|<|\mathcal{P}(\omega)|$ in the sense that there exists injective $f:\mathcal{P}(\omega)\rightarrow\mathcal{B}$ but not the other way round. In fact, Hjorth shows in his paper An Absoluteness Principle for Borel Sets that under determinacy, if $\alpha<\beta<\omega_1$ then there is no injection from $\mathbf{\Pi_\beta}$ (the $\beta$-th level of Borel hierarchy) to $\mathbf{\Pi_\alpha}$.