I am reading Richard Kaye's Models of Peano Arithmetic. In chapter 7 we define the collection axioms $B\varphi$ for $\mathscr L_A$-formulas $\varphi$ which are, to quote, 'the arithmetic analogue of Fraenkel's collection principle in set theory'. I could not find other sources which mention them, so I copied the definition below.
What is the motivation for this definition? Is there a way to simplify the definition of the theory $B\Sigma_n$ in words?
Thank you!
Let $\varphi(x,\overline y,\overline z)$ be a formula of $\mathscr{L}_A$. The collection axiom for $\varphi$ is the sentence $B\varphi$, which is $$\forall\overline z,t\big(\forall x<t\;\exists \overline y\;\varphi(x,\overline y,\overline z)\rightarrow \exists s\;\forall x<t\;\exists\overline y<s\;\varphi(x,\overline y,\overline z)\big).\tag{1}$$ Then $B\Sigma_n$ is the theory $$I\Delta_0+PA^-+\{B\varphi:\varphi\;\text{is a}\;\Sigma_n\;\mathscr{L}_A\text{-formula}\}.$$
$(1)$ seems to say that whenever for certain $t$s we can find arbitrary $\overline y$s to satisfy $\varphi$, then we can do it for $\overline y$s bounded by a certain $s$. In passing it is remarked that the collection axioms can be used to show that the classes $\Sigma_1$ and $\Pi_1$ are closed under bounded quantification—which I take to mean that if $\varphi$ is $\Sigma_1$ (or $\Pi_1$), then for any $x$, $\exists x<a\varphi$ (or $\forall\dots$) is also $\Sigma_1$ (or $\Pi_1$)—though this is not obvious to me.