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.



Some "axiom of choice" and "dependent choice" issues
To elaborate a bit on Ashutosh's comment, [my answer to this question](https://mathoverflow.net/a/32722/1946) 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. 


