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One of the known ways to axiomatize ZF is via these two axiom schemata:

Separation: $\forall \vec{w} \forall A \exists! x: x=\{y \in A: \phi\}$

Reflection: $\forall \vec{w} \exists \alpha: \phi \to \phi^{V_\alpha}$

Now if we add to this a primitive unary function symbol $j$, and stipulate that $j$ is nontrivial elementary embedding of the whole universe to itself, but forbid the use of $j$ in separation and permit it in Reflection.

What would be the effect of that?

Would the resulting theory be consistent with choice?

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    $\begingroup$ Regarding your second question, if $j:L\to L$ is elementary (where say $j$ is definable) then $(L,j)$ models the theory plus choice. $\endgroup$
    – Farmer S
    Commented Feb 24, 2021 at 20:48
  • $\begingroup$ @FarmerS, but why should such $j$ exist. $\endgroup$
    – Zuhair Al-Johar
    Commented Feb 25, 2021 at 10:15
  • $\begingroup$ I just mean assume there is one. See $0^\#$ (zero sharp). $\endgroup$
    – Farmer S
    Commented Feb 25, 2021 at 11:58
  • $\begingroup$ Or suppose $k:V\to M$ is elementary, $M$ transitive; then $L^M=L$, so the restriction $j$ of $k$ to $L$ is elementary $L\to L$. I also wanted $j$ to be definable, which can be arranged with more work. But that doesn't really matter; we just need $j$ a class, which follows if $k$ is. $\endgroup$
    – Farmer S
    Commented Feb 25, 2021 at 12:03

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