In [posting](https://mathoverflow.net/questions/317658/what-is-the-strength-of-adding-limitation-of-size-and-a-simple-version-of-reflec) 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?



What is the limit to iterating class comprehension, reflection and limitation of size?
I claim $K^+(V^\lambda)$, for any $\lambda$ (I switched the notation so not to be confused with the von Neumann universe) is equiconsistent with the schema "$ORD$ is Mahlo" (Not to be confused with full stationarity, which could be called "$Ord$ is Mahlo" if you really wanted to distinguish them). First off each $V^\lambda$ is a Grothendieck universe, and so of the form $V\_\kappa$ for inaccessible $\kappa$. Second of all, $V\_\kappa\prec W$, where $W=\{x|x=x\}$. To see this, suppose $\exists x(\phi(x,x\_0...x\_n))$ where $\phi(x,x\_0...x\_n)$ is absolute.


Then $\exists x(\phi(x,x\_0...x\_n))$ if and only if $\exists x\in V^\lambda(\phi(x,x\_0...x\_n))$ if and only if $\exists x\in V^\lambda(\phi^{V^\lambda}(x,x\_0...x\_n))$ if and only if $V^\lambda\vDash\exists x(\phi(x,x\_0...x\_n))$. Note that at no point do we use reflection for a formula that uses $V^\lambda$. Therefore $K^+(V^\lambda)$ proves that there exists a reflection cardinal, and so the consistency strength of $K^+(V^\lambda)\ge$ the consistency strength of "$ORD$ is Mahlo."


Then, suppose $ORD$ is Mahlo. Then there exists a proper class of reflecting cardinals, and if we take $V^\lambda=V\_\kappa$ for reflecting $\kappa$, we get $K^+(V^\lambda)$. Therefore the consistency strength of "$ORD$ is Mahlo"$\ge$ the consistency strength of $K^+(V^\lambda)$, and so the consistency of "$ORD$ is Mahlo"$=$ the consistency strength of $K^+(V^\lambda)$.


