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This question is related to a question lately posted to $\cal MO$. Here, we add two partial unary functions $``j,f"$ to the language of $\sf ZF$.

The question is about if we can add the following on top of axioms of $\sf ZF$ [$j,f$ not used in Replacement nor Separation]?

$\exists \alpha: \text{ limit} (\alpha) \land j: V_\alpha \to V_{\alpha +1} \land f: V_\alpha \to V_\alpha \land j,f \text{ are bijections } \\ \forall S: j[S]; j^{-1} [S] \text { both exist } \\ \forall S \in V_\alpha: j(S)=f[S] $

Where: $g[S]=\{g(x) \mid x \in S\}$

The first two conditions have already been proved consistent, it's the addition of the last condtion that is unsolved?

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The above theory is inconsistent. Since j is surjective, there is an xโˆˆ๐‘‰๐›ผ with jx=๐‘‰๐›ผ. Then f[x]=๐‘‰๐›ผ. Therefore x=๐‘‰๐›ผ, since f is a bijection. But ๐‘‰๐›ผ is not an element of ๐‘‰๐›ผ.

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