When publishing on a principle of reflection together with a limitation of the size of the axiom on the theory of sets of Ackermann, the answer is that the theory rises to a cardinal of Mahlo.

I'm here wondering if this method can be iterated, and what is the most it can achieve through this iteration process.

For example, let's define a theory. $ mathsf {K} ^ {+} (V _ { lambda}) $ in the language of $ FOL (=, in, V_1, V_2, .., V _ { lambda}) $ While $ lambda $ is a specific recursive ordinal that has some specific ordinal notation, that is, whenever $ lambda < omega_1 ^ {CK} $

Now the idea is that every theory $ mathsf {K} ^ {+} (V _ { lambda}) $ has axioms of extensionality, class understanding axiom scheme for $ V _ { alpha} $, an axiom of reflection for $ V _ { alpha} $, and axiom size limitation for $ V _ { alpha} $, for each $ alpha < lambda $, we also have the axiom scheme:

Yes $ alpha < beta $, so: $ “ forall x (x subset V _ { alpha} to x in V { beta}) "$

it is an axiom

More specifically the formula of class understanding for $ V _ { alpha} $ is:

$$ for all x_1, .., x_n subseteq V _ { alpha} exists x forally (y in x leftrightarrow and in V { alpha wedge varphi (y, x_1,. ., x_n)) $$, where $ varphi (y, x_1, .., x_n) $ It is a formula that does not use primitives. $ V _ { beta} $ when $ beta> alpha $.

While the formula of the reflection scheme for $ V _ { alpha} $ it 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) $ does not use any primitive symbol $ V _ { beta} $ While $ beta geq alpha $.

Now, what is the limit of the strength of the force? $ mathsf {K} ^ {+} (V _ { lambda}) $ theories