# Definable $\Sigma_1$-elementary submodels

Given $K \models ZFC-P$ transitive and $M\prec K$ countable, suppose $W\prec_{\Sigma_1} K$ is unique with respect to some first order property $\psi(W,z)$ where $z\in K$ is a parameter.

Now if $z\in M$, then claim that $W\in M$. To see this, consider the sentence: $\sigma(z):\equiv \exists ! W \psi(W,z) \ \& \ \forall \bar{a}\in W^{<\omega} \forall \varphi(\bar{x}, y)\in Formula_{\Sigma_1} \ \models^{\Sigma_1} \exists y \varphi(\bar{a},y) \rightarrow \exists y\in W \varphi(\bar{a},y)$.

If $\sigma$ is a first-order sentence, then we are done by elementarity. The only place that I'm not 100% sure is that $\models^{\Sigma_1}$ is actually first order definable (in fact I need it to be sort of uniformly definable). I believe this is also the point for considering $\Sigma_1$ elementary submodels as if $W\prec K$ there is no reason to believe that $W\in M$ since $\models^{First\ order}$ is not first order definable. Am I right? Thanks!

• I assume you want $K$ to be transitive? – Asaf Karagila Jul 22 '17 at 19:12
• @AsafKaragila oh yes – Otto Jul 22 '17 at 19:12