User blog comment:PsiCubed2/Question: General breakdown of levels up to Second Order Arithmetic/@comment-1605058-20180203222427

1. You will likely be interested in Madore's ordinal zoo. Noteworthy point is 2.14, the nonprojectible ordinal, which corresponds to the strength of \(\Pi^1_2-CA_0\), and 2.17, which corresponds to full Z2.

2, 3 & 4. "Reflection" in general means results of the sort "if something holds in the universe, then it holds in some set". More precisely, \(\Pi_n\)-reflection is the following axiom schema: for each \(\Pi_n\) formula \(F\), "\(F\implies\exists z:z\) is a transitive set and \(F\) holds in \(z\)" (we also allow parameters, and then they have to be in \(z\)). However, for many purposes, we want to assume not only that \(z\) is transitive, but also that \(z\) is admissible, i.e. satisfies KP. For small \(n\) this significantly strengthens all the reflection principles (non-admissible \(\Pi_2\)-reflection is provable in KP), but I believe it makes no difference for \(n>2\). We call an admissible ordinal \(\alpha\) \(\Pi_n\)-reflecting if this stronger reflection holds in \(L_\alpha\).

\(\Pi_1\)-reflection is a pretty trivial matter - a \(\Pi_1\) formula says that something holds for every set, so it will surely hold in every set \(z\) as well. Admissible \(\Pi_1\)-reflection, then, basically says that every set is contained in an admissible set. Hence an admissible ordinal \(\alpha\) is \(\Pi_1\)-reflecting iff admissible ordinals are unbounded in \(\alpha\), i.e. it's recursively inaccessible.

Similarly, it can be shown that \(\Pi_2\)-reflecting ordinals are the recursively Mahlo ordinals (to see why this is so, we can view true \(\Pi_2\) formulas, which have the form \(\forall x\exists y:\), as functions assigning some \(y\) to each \(x\), and then an admissible set satisfying the reflection instance will be a fixed point, which can be related to an admissible ordinal lying in the set of values of the function, and that's roughly the definition of recursively Mahlo).

Hence, using reflection principles with admissibles, \(\Pi_1\)-reflection corresponds to (recursive) inaccessibility and \(\Pi_2\)-reflection corresponds to (recursive) Mahloness (and using reflection without admissibles makes them basically trivial).