While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then wondered whether this was unique. When I got back to my computer, I looked on google and came up with this, which indicated that is not. However, the proof given there makes heavy use of Choice, and based on similar things, I am guessing that this cannot be avoided.
Define DC($\omega_1$) as:
For all trees $T$, $\quad$ $T\:$ has a branch $\ $ or $\ $ $T\:$ has a chain of length $\omega_1$ $\quad$.
which, if I haven't messed up the simplification, ZF proves is equivalent to the definition given at shelah.logic.at/files/446.ps .
Does $\ ZF+DC(\omega_1)\ $ prove that there is a unique subfield of $\mathbb{C}$ which is both isomorphic to $\mathbb{R}$ and has Baire property?
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