# Is it true that $V$ and $H_{\omega_1}$ agree on the truth value of $\Sigma_1$ sentences?

I want to see if the following result is correct or not:

Let $\varphi(x)$ be a formula in $\mathcal{L}_{\mathrm{ZF}}$ with only bounded quantifiers such that $\exists x\,\varphi(x)$ holds in $V$, then $\exists x\,\varphi(x)$ holds in $H_{\omega_1}$.

I wrote a proof for it using the following reasoning:

$\varphi$ is absolute for all transitive models; let $\exists x\,\varphi(x)$ be true, then $\varphi(a)$ holds for some $a$. Let $\vartheta\in\mathbf{ON}$ such that $a\in V_\vartheta$. Let $M\prec V_\vartheta$ be a countable elementary substructure, $\bar M$ be the Mostowski collapse ($\bar M\cong M$), then $\bar M$ is transitive and countable, hence hereditarily countable, so $\bar M\subseteq H_{\omega_1}$, so there is a witness for $\varphi(x)$ in $H_{\omega_1}$.

My questions are: (1) is the result above correct? (2) Is the proof given correct? (3) If the result is correct and the proof is not correct, what would be a correct proof for the result?

• The result and your proof are correct. But your result can be strengthened to let $\phi$ contain parameters from $H_{\omega_1}$
– user104955
Mar 9, 2016 at 21:08
• @GME Thank you, and if $\varphi$ contains parameters from $H_\kappa$ for $\kappa$ uncountable then I can show absoluteness of $\varphi$ for $H_\kappa$?
– Marc
Mar 9, 2016 at 21:10
• Yes, that's right!
– user104955
Mar 9, 2016 at 21:17

We assume that $$\exists x\varphi(x)$$ is true in $$V$$, so there is some $$a$$ such that $$\varphi(a)$$ holds. Let $$\vartheta$$ an ordinal such that $$a\in V_\vartheta$$, since $$V_\vartheta$$ is transitive and $$\varphi$$ is a bounded formula, $$V_\vartheta\models\varphi(a)$$ and therefore $$V_\vartheta\models\exists x\varphi(x)$$. Let $$M$$ be a countable elementary submodel of $$V_\vartheta$$ and let $$\overline M$$ be its Mostowski collapse, then $$\overline M\in H_{\omega_1}$$.
Finally, since $$M$$ is an elementary submodel of $$V_\vartheta$$, and $$M\cong\overline M$$ we have that $$\overline M\models\exists x\varphi(x)$$. So there is some $$a'\in\overline M$$ such that $$\overline M\models\varphi(a')$$, but again by absoluteness of bounded formulas between transitive sets, $$H_{\omega_1}\models\varphi(a')$$ so $$H_{\omega_1}\models\exists x\varphi(x)$$ as wanted.
Essentially the same proof can be extended to show that $$H_\kappa\prec_{\Sigma_1}V$$, as remarked by GME in the comments.