I'll largely present this posting informally, because I'm not certain if it can be done formally in the language first order logic with identity and membership.
What I want to do is to iterate reflection, the form I'm concerned with is that following:
$$\forall \vec{w} \ \exists \alpha: \varphi(\alpha) \land [\psi \to \psi^{V_\alpha}]$$
Now the idea is change $\varphi$, so we are to have for example formulas $\varphi_0,\varphi_1, \varphi_2,...$, now each formula $\varphi_{i+1}$ is to be defined in terms of $\varphi_i$, more specifically I want $\varphi_{i+1}$ to say that $\alpha$ is a rank of a universe of a theory having axioms: Extensionality, Separation, and reflection...up to $\varphi_i$, the latter is all instances of the following up to $n=i$: $$\forall \vec{w} \ \exists \alpha: \varphi_n(\alpha) \land [\psi \to \psi^{V_\alpha}]$$ for example we begin with $\varphi_0$ to be "is ordinal", now I expect $\varphi_1$ to be "is inaccessible", while $\varphi_2$ to be "is hyper-inaccessible", etc...
Can this be formalized in first order logic?
If this can be done, can we extend the indexing of the $\varphi$'s [through using ordinal notations] as to include all recursive ordinals (i.e. ordinals below $\omega_1^ {CK})$? And in this case how can one define the $\varphi_i$'s when $i$ is a limit?
If the above can be done, what would be the strength of this iterative reflection?