Why we care only about $\Pi_1$ parameters in axiom schemes?

There is exist tradition to formulate axiom schemes such as induction, comprehension or replacment in a way like "for each formula $$\phi(y,\overline{x})$$ with free variables $$y$$ and $$\overline{x}=x_1,...,x_n$$ we have $$\forall \overline{x} ( \text{text of axiom scheme which include } \phi(y,\overline x))$$". To not be vague, let take as archetypical example comprehension scheme for second order arithmetics from ncatlab article.

$$\forall \overline m \forall \overline X \exists Z \forall n (n \in Z \leftrightarrow \phi(n,\overline m, \overline X))$$

I choose that example because I don't know it is equivalent to axiom without parameters or not, whenever equivalence to $$ZFC$$ or first order $$PA$$ to versions without parameters is classical result. I understand that apriori allowing parameters could increase expressible power of our system, but I don't understand why we not allowing not only $$\Pi_1$$ parameters but parameters of all type like, for example, next $$\Sigma_3$$ axiom scheme

$$\exists \overline m_1 \exists \overline X_1 \forall \overline m_2 \forall \overline X_2 \exists \overline m_3 \exists \overline X_3 \exists Z \forall n (n \in Z \leftrightarrow \phi(n,\overline m_1, \overline X_1, \overline m_2, \overline X_2, \overline m_3, \overline X_3))$$

it seems for me like it could increase our expressible power a lot, so, why it is not traditional?

• @MauroALLEGRANZA You are right it was mistyping. I am fix it – kp9r4d Mar 13 at 12:33
• What does it mean for a parameter to be $\Pi_1$? – Asaf Karagila Mar 13 at 12:45
• I'm not sure to understand what kind of "flexibility" are you thinking about... Consider the simple case $∀m∃Z∀n(n∈Z ↔ ϕ(n,m))$; the schema means that, for fixed $\phi$, we have a "collections" of sets $Z_m$, one for each value of the parameter $m$. – Mauro ALLEGRANZA Mar 13 at 12:57
• If we instead use $∃$ what we get is $∃m∃Z∀n(n∈Z ↔ ϕ(n,m))$ that means that - for the specific value of $m$ that satisfies the formula - we have a single set $Z$. – Mauro ALLEGRANZA Mar 13 at 12:58
• @AsafKaragila It means that it have form $\forall \text{param} (\text{axiom scheme})$ as analogy to arithmetical hierarchy of formulas. – kp9r4d Mar 13 at 13:00