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$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$.

Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of $L_Q$. If $A$ is an $L_Q$-structure, then $A\models QxP$ if and only if $|\{a\in A\mid A\models P[a]\}|>\aleph_0$.

Which structure could be model of the formula that is the conjunction of the axioms that say that $R$ is a total order and $Qx(x=x) \land \forall x\lnot Qy(yRx)$

In the real numbers?...but $\mathbb R$??

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You should probably give the definition of $Q$ (or at least point to your previous post where the definition is given). – Asaf Karagila Jan 31 '13 at 10:41

The real numbers clearly cannot satisfy this formula, because it implies that every initial segment is countable.

On the other hand, consider the ordinal $\omega_1$, it is an uncountable linear order and every initial segment is countable.

Similar models would be replacing every point in $\omega_1$ with any countable linear order, and I believe that you can show that this is really all the models of this theory.

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