Can we upgrade the Reflection axiom schema in Ackermann to the following:

Modified Reflection axiom schema: if $\psi(y)$ is a formula that doesn't use the symbol $V$, in which only symbols $y,x_1,..,x_n$ occur free, and in which $x$ is not free, then all closures of:

$$x_1,..,x_n \in V \wedge \forall y (\psi(y) \to y \subset V) \to \\\exists x \in V \forall y (y \in x \leftrightarrow \psi(y))$$; are axioms.

In the traditional exposition of Ackermann set theory, the output of $\psi(y)$ is restricted to elements of $V$. While here it is eased as to allow subsets of $V$.

With the above formulation, there is no need for a second completeness axiom for $V$, since it would be redundant. We only need the axiom of heredity, that is the first completeness axiom for $V$, which only mounts to saying that the world $V$ of all sets is transitive, a very natural statement!

Question: Is there a clear problem with the above scheme?

I tend to think that the above schema is a theorem schema of Ackermann set theory.


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