# A problem with a $\Pi_1$ formula of the Lévy hierarchy

Let $$M\equiv N$$ means that $$(M,\in_M)$$ and $$(N,\in_N)$$ satisfy the same sentences of the language of set theory, with $$\in_M$$ and $$\in_N$$ being the standard membership relation restricted to $$M\times M$$ and $$N\times N$$, respectively.

My question is about what level of Lévy hierarchy the formula $$\Psi(\alpha,\beta):= (V_\alpha \equiv V_\beta)$$ is in. At first, I thought that was in $$\Pi_1$$, but after some reflection, I see it is wrong.

This statement has complexity $$\Delta_2$$, because it is a locally verifiable feature, meaning one that can be decided yes-or-no inside any sufficiently large $$V_\theta$$. Any $$V_\theta$$ above $$\alpha$$ and $$\beta$$ will correctly determine whether $$V_\alpha\equiv V_\beta$$.
Let me add that most of the complexity of $$V_\alpha\equiv V_\beta$$, however, as a relation on $$\alpha$$ and $$\beta$$, is just knowing that you have the right $$V_\alpha$$ and $$V_\beta$$. In contrast, the relation $$M\equiv N$$, as a relation on $$M$$ and $$N$$, is $$\Delta_1$$, since you just have to say that the unique satisfaction relations for these structures fulfill the same sentences. Because the satisfaction relations are unique in fulfilling a certain $$\Delta_0$$ expressible recursion, we can express this in a $$\Sigma_1$$ way or a $$\Pi_1$$ way, making them $$\Delta_1$$. Consequently, your $$\Pi_1$$ case would be fine if we view it as a relation on pairs $$(V_\alpha,V_\beta)$$, rather than as a relation on $$(\alpha,\beta)$$.
One can see that $$V_\alpha\equiv V_\beta$$ is not complexity $$\Pi_1$$ in $$(\alpha,\beta)$$ in all models of set theory by observing that it is not always downwards absolute. For example, perform an Easton-support product to form an extension $$V[G]$$ by forcing a violation of GCH at every regular cardinal. Now by cardinality considerations there will be ordinals $$\alpha\neq \beta$$ such that $$V[G]_\alpha$$ and $$V[G]_\beta$$ have the same theory, where $$\alpha=\kappa+3$$ for some regular cardinal $$\kappa$$. If $$V[G^-]\subseteq V[G]$$ is the submodel that omits the forcing at coordinate $$\kappa$$, then $$V[G^-]_\alpha$$ is not elementarily equivalent to $$V[G^-]_\beta$$, since the former knows that the missing coordinate is very near the top, but the latter doesn't.
• Answome answer professor. This argument implies that $V_\alpha\prec V_\beta$ is $\ \Delta_2$ too? I wil love if you can indicate material to I learn more about these topics Apr 14 at 21:16