# What is the limit to iterating class comprehension, reflection and limitation of size?

In posting about a reflection principle coupled with a limitation of size axiom over Ackermann set theory, the answer is that the theory is blown up to a Mahlo cardinal.

I'm here just wondering if this method can be iterated, and what is the maximal that it can reach to via this iteration process.

For example lets define a theory $$\mathsf{K}^{+}(V_{\lambda})$$ in the language of $$FOL(=,\in, V_1, V_2,..,V_{\lambda})$$ as long as $$\lambda$$ is some specific recursive ordinal having some specific ordinal notation, i.e. as long as $$\lambda < \omega_1^{CK}$$

Now the idea is that each theory $$\mathsf{K}^{+}(V_{\lambda})$$ has axioms of Extensionality, Class comprehension axiom schema for $$V_{\alpha}$$, a reflection axiom scheme for $$V_{\alpha}$$, and limitation of size axiom for $$V_{\alpha}$$, for each $$\alpha < \lambda$$, also we have the axiom schema:

if $$\alpha < \beta$$, then: $$\forall x (x \subset V_{\alpha} \to x \in V_{\beta})"$$ is an axiom.

More specifically the formula of class comprehension for $$V_{\alpha}$$ is:

$$\forall x_1,..,x_n \subseteq V_{\alpha} \exists x \forall y (y \in x \leftrightarrow y \in V_{\alpha} \wedge \varphi(y,x_1,..,x_n))$$, where $$\varphi(y,x_1,..,x_n)$$ is a formula that do not use primitives $$V_{\beta}$$ when $$\beta>\alpha$$.

While the formula of reflection schema for $$V_{\alpha}$$ would be written as:

$$\forall x_1,..,x_n \in V_{\alpha} \\ [\exists y (\varphi(y,x_1,..,x_n)) \to \exists y \in V_{\alpha}(\varphi(y,x_1,..,x_n)) ]$$ where $$\varphi(y,x_1,..,x_n)$$ doesn't use any primitive symbol $$V_{\beta}$$ as long as $$\beta \geq \alpha$$.

Now what is the limit to the consistency strength of the $$\mathsf{K}^{+}(V_{\lambda})$$ theories?