Elementary submodels of $(V_{\omega},\in)$ are equal to it

Question

I read that all the ESMs of $(V_{\omega},\in)$ will be equal to it . But what if the ESM's universe is a finite set?

Answer 1

HINT: If $M\prec N$, and $x\in N$ is a definable element, then $x\in M$. Prove, by induction, that every element of $V_\omega$ is definable.