# About automorphism over non-well-founded models of $\sf ZF+V= L(A)$? Answer

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## About automorphism over non-well-founded models of $\sf ZF+V= L(A)$?

Can we have a set $$A$$ and a non-well founded model $$M$$ of $$\sf ZF + V=L(A)$$ such that there exists an external automorphism $$j$$ over $$M$$ and a non-standard limit ordinal $$\alpha \in M$$ other $$j(L_\alpha(A)) =L_\beta(A)$$ other $$|\alpha|=|\beta|^+$$ ?

Where $$L(A)$$ is the relatively constructible hierarchy raised over set $$A$$

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