To simplify the notation, assume $V=L$. We have $\lvert V\_{\omega\_{1}} \rvert=\aleph\_{\omega\_{1}}$ and $\lvert H(\aleph\_{1})\rvert=\aleph\_{1}$, so in particular $V\_{\omega\_{1}} \models \exists x \forall \alpha\; x \not\in L(\alpha) $ since $L(\omega\_{1})=H(\aleph\_{1})$.


Using the Löwenheim–Skolem theorem we have a transitive countable set $M\prec V\_{\omega\_{1}}$. In particular, $M\in H(\aleph\_{1})$. We have $H(\aleph\_{1})\prec\_{1}V\_{\omega\_{1}}$, so $M\prec\_{1}H(\aleph\_{1})$. By the condensation lemma $M=L(\alpha)$. $\alpha$ must be a limit ordinal, but $L(\beta)\in L(\beta+1)$, so $L(\alpha)\models \forall x\exists \beta \; x\in L(\beta)$ for all ordinal limit $\alpha$. This is a contradiction with $M\prec V\_{\omega\_{1}}$.


Where is the mistake? I didn't find it.


Edit: I think the Mostowski collapse lemma is not the core of problem.
We have that exist a $\alpha$ limit with $V\_{\omega\_{1}}\in L(\alpha)$, and we have $V\_{\omega\_{1}}\prec\_{1}L(\alpha)$, but as $V\_{\omega\_{1}}$ is transitive his transitive collapse is himself, but is not possible.


The heart of problem is the relation $V\_{\omega\_{1}}\prec\_{1} L(\alpha)$, but why is it wrong?


Edit: thanks the comentaries the solution is, despite $V\_{\omega\_{1}}$ and $L(\alpha)$ satisfies the same $\Sigma\_{1}$ sentences $\varphi$, we need to consider all formulas $\varphi(x)$ that are $\Sigma\_{1}$



A doubt about the Gödel condensation lemma
Your mistake is that taking the Mostowski collapse does **not** preserve elementarity.


We do have a countable transitive $A$ and a countable $M$ with $$A\cong M\preccurlyeq V\_{\omega\_1},$$ where $M$ comes from downward Lowenheim-Skolem applied to $V\_{\omega\_1}$ and $A$ is the collapse of $M$, but that does not imply $A\preccurlyeq V\_{\omega\_1}$. Elementary substructurehood is more than just agreement of theories, it also takes into account exactly how the smaller structure sits inside the larger one. Moving the smaller structure around (e.g. via the Mostowski collapse) may break elementarity.


