Chang and Keisler's Model theory gives the following exercise problem:
Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models.
(A model $\mathfrak{A}$ of set theory is natural if $\mathfrak{A}=(V_\alpha,\in)$ for some $\alpha$. )
I don't know how to prove this problem. Any hints or ideas will be very appreciated.
Here's a simple way to do it. Suppose it fails. Then let $F(T)$ be the least ordinal $\alpha$ such that for all $\beta>\alpha$, $V_\beta\not\vDash T$ (where $T$ is a complete extension of $Z$). Since the complete extensions of $Z$ form a set, Replacement implies that $rng(F)$ is a set and thus that it has a least upper bound $\beta$. Let $\lambda$ be the least limit greater than $\beta$. Then $V_\lambda\not\vDash T$ for all complete extensions $T$ of $Z$, which is impossible.