This question takes place in ZF. Assume some mild large cardinals; then it is consistent (in fact, it follows from AD, the consistency of which follows from mild large cardinals) that there are very large well-ordered cardinals onto which $\mathbb{R}$ surjects (cf. https://mathoverflow.net/questions/47028/value-of-theta-in-zfad). I am interested in the converse: what sorts of sets can inject into $\mathbb{R}$?
My main question is, Is it consistent with ZF that there is some measurable well-ordered cardinal $\mu$ and an injection $i: \mu\rightarrow\mathbb{R}$? I suspect that this question has an easy negative answer which I'm just not seeing right now, hence my asking here as opposed to at the Overflow.
A secondary, and vaguer, question, is the following: in ZF (or ZF+AD), what sorts of sets can inject into $\mathbb{R}$?
Thank you all very much in advance!