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.

7more comments