See this question on Mathoverflow.
Can we have this with stratified Zermelo. Where the latter is Zermelo set theory but with separation restricted to stratified formulas only. To present the question in full, I'll re-iterate it here:
Define a pre-ordinal as a transitive set of transitive sets.
Is it consistent with Stratified Zermelo set theory (without choice) to have a nonempty set $S$ such that: for every element $s \in S$ there exists a pre-ordinal $\beta$ and a set $Y$ such that $s=\langle \beta, Y \rangle $, and there exists an element $r \in S$ such that: $r= \langle \alpha, X \rangle$, where $\beta=\alpha \cup \{\alpha\}$ and $Y = \cal P$$(X)\backslash \{\emptyset\}$.