I'm trying to understand how the Levy hierarchy does not contradict the axiom schema of restricted separation.
In his paper on the set hierarchy Levy introduced satisfaction predicates which I think implies that for any fixed quantifier rank $k$, there is a predicate $\phi_k$ of the same rank with one additional free variable $x$ such that $\phi_k(x)$ is true precisely when the formula of rank $k$ with Godel number $x$ is also (assuming $x\in V_\omega$ is part of a Godel numbering scheme). Also, since such predicates deal with all formulas of a fixed rank they are not equivalent to formulas of a lesser rank.
Assuming I got that right, then I don't understand how we can have the separation axioms be consistent because we would have: $$\exists b\; \forall x\; x \in b \leftrightarrow \phi_k(x) \wedge x \in V_\omega$$ but the left side of the bi-conditional is just set membership, a formula of rank $0$, so how could it be equivalent to a formula of a much higher rank? Is the issue that I need to account for the set $b$ not being part of the language itself but rather the universe of a model? Thanks.