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The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement

$\varphi$ has a model

absolute between $V$ and $W$? It is clearly upwards absolute, by induction on rank, so I'm really asking about the downwards direction.

The main obstacle that I see right now is the lack of a Lowenheim-Skolem theorem, which prevents us from using Shoenfield absoluteness. As a specific example, let $A$ be $\omega_1^W$ as a linear order, and suppose $V$ is a forcing extension of $W$ in which $\omega_1$ is collapsed. In $V$, there is a sentence $\varphi\in\mathcal{L}_{\omega_1\omega}\subset\mathcal{L}_{\infty\omega}$ whose only countable model up to isomoprhism is $A$: the Scott sentence of $A$. Now there is a formula $\psi\in \mathcal{L}_{\infty\omega}\cap W$ such that, in $V$, $\psi$ and $\varphi$ are equivalent; this is a roundabout way of saying "$\varphi\in W$," which is not exactly true. But then $\psi$ has no countable models in $W$, since such a model would have to be isomorphic to $A$ in $V$.

I assume this is well-known, but I haven't been able to find an answer myself.

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Here is a somewhat easier counterexample.

Let $I$ be a countable set, which is uncountable in $W$. Let $c_n$ and $d_\alpha$ be constant symbols, for $n\in\mathbb{N}$ and $\alpha\in I$. Consider the formula $\varphi$ that asserts that all the $d_\alpha$'s are different, but that every $d_\alpha$ is equal to some $c_n$. That is, $$\varphi=\bigl(\bigwedge_{\alpha\neq\beta\in I}d_\alpha\neq d_\beta\bigr)\wedge\bigl(\bigwedge_\alpha\bigvee_n d_\alpha=c_n\bigr).$$ A model of this sentence is essentially providing an injective function from $I$ to $\mathbb{N}$, mapping $\alpha\mapsto n$ when $n$ is least for which $d_\alpha=c_n$. Thus, the sentence is not satisfiable in $W$, since $I$ is uncountable there, but it is satisfiable in $V$, since $I$ is countable in $V$.

Using the same idea, one can make a counterexample in a finite signature language as follows. Suppose that $P(\mathbb{N})^W$ is countable in $V$. First, we can make a sentence that asserts that we have a copy of $\langle\mathbb{N},S\rangle$, by asserting that everything in that sort is a finite successor of $0$. Next, we can have another sort that will be a copy of $P(\mathbb{N})^W$, each subset represented by a vertex pointing exactly at its elements. In a single formula (of size continuum in $W$), we can say that every subset of $\mathbb{N}$ that it is represented by some vertex. Finally, with one more binary relation, we can say that every subset is associated with a distinct natural number. The point now is that the overall assertion is not satisfiable in $W$, since $P(\mathbb{N})^W$ is uncountable in $W$, but it will be satisfiable in $V$, since the set is countable there.

Update. Meanwhile, the satisfiability of sentences in $\cal{L}_{\omega_1,\omega}$ logic, that is, using only countable meets and joins, is a $\Sigma^1_1$ assertion and hence absolute between models with the same countable ordinals.

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  • $\begingroup$ This is lovely, and very simple - thanks! $\endgroup$ – Noah Schweber Oct 23 '13 at 2:27
  • $\begingroup$ With a bit of work, this can even be turned into a complete $\mathcal{L}_{\infty\omega}$ sentence, as well. $\endgroup$ – Noah Schweber Oct 23 '13 at 3:25
  • $\begingroup$ This is a comment on JDH's "Update" rather than the original question: For $L_{\omega_1,\omega}$ sentences, model existence in $\aleph_0$ and $\aleph_1$ is absolute. $\endgroup$ – Ioannis Souldatos Jul 28 '15 at 19:45

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