User blog comment:PsiCubed2/Question: General breakdown of levels up to Second Order Arithmetic/@comment-30118230-20180204192313

LittlePeng9 gave you an exelent explanation for 2,3 & 4 and as for your first question,I will add this to simplify the answer:

The general list of "milestones" towards Z2 consists of $$\Pi^1_1-CA_0,\Pi^1_2-CA_0,\Pi^1_3-CA_0,.........,\Pi^1_n-CA_0,......$$ and the limit (more precisely union) of these gives you $$\Pi^1_\infty-CA$$,also (more commonly) known as $$Z_2$$.

These "accelerate" quite rapidly,so here is a list of "milestones" for just $$\Pi^1_1-CA_0$$: RCA0, WKL0, ACA0, ATR0; The next set of "milestones"to $$\Pi^1_2-CA_0$$ mainly consists of $$\Pi^1_1-CA_0,\Pi^1_1-CA,\Pi^1_1-CA+BI,\Delta^1_2-CA,\Pi^1_1-TR_0,\Delta^1_2-CA+BI,......$$ ect. this list has no pattern to it whatsoever,but it's a nice way to look at little steps towards $$\Pi^1_2-CA_0$$,even though $$\Pi^1_2-CA_0$$ is already very,very strong. It goes far past standard OCFs and is quite difficult to analyse. That's why it's PTO (in terms of an assigned ordinal notation made specifically as a model for it) still hasn't quite been diescovered,except for a few papers by Rathjen that talk about it,but not in great detail.

Each $$\Pi^1_n-CA_0$$ refers to a new type of formula combining the comprhension axiom and $$\Pi^1_n-F$$ generally give you an extra axiom F combined with a $$\Pi^1_n$$ type formula.