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Yes. Just take any countable $\omega$-nonstandard model $M\models\text{ZFC}$. In this case, once $\alpha$ is above the standard finite numbers, we can biject $V_{\alpha+2}^M-V_{\alpha+1}^M$ with $V_{\alpha+1}^M-V_\alpha^M$, since these will both be countably infinite sets. Putting these all together, and fixing the standard hereditarily finite sets, we get a permutation $j$ of $M$ that sends $V_{\alpha+1}^M$ exactly to $V_\alpha^M$ for every infinite ordinal $\alpha$ (as well as for all the nonstandard finite numbers $\alpha$).