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Is there an $\mathcal{L}_{\omega_1\omega}$ formula or set of formulas that characterizes the rationals $\mathbb{Q}$ among other number fields?

EDIT: My formula must not contain an infinite number of constants from $\mathbb{Q}$.

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What's wrong with the obvious formula $\forall x . \bigvee_{a \in \mathbb{Q}} x = a$? –  Zhen Lin Sep 19 '12 at 9:40
    
I added one further restriction. See Edit. Sorry about that. –  user16974 Sep 19 '12 at 9:53
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Well, one can still seem to do it in the language of fields with $\forall x. \bigvee_{n,m\in\mathbb{N}} x= \pm(1+\cdots+1)/(1+\cdots+1)$, plus $1+\cdots+1\neq 0$. I am not using $n$ and $m$ as parameters, but rather just to specify syntactically the number of $1$'s in the expression. –  Joel David Hamkins Sep 19 '12 at 10:58
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