= Rank-shifting automorphism construction of NFU
In a model of <ZFC>, choose an external automorphism $j$ and an ordinal $\alpha$ with $j(\alpha)>\alpha$. Set $D_i=V_{j^i(\alpha)}$ for integer $i$, and declare the sets of type $i+1$ to be its objects that are subsets of $D_i$. Ordinary membership between successive types satisfies typed comprehension and set-extensionality. The automorphism shifts all types, yielding <NFU> by the type-zero collapse. A <Skolem hull> of integer-indexed increasing ordinal indiscernibles supplies such a $j$ by shifting the indices. The model is externally ill-founded; a transitive well-founded universe cannot have such an automorphism.
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