We work in a countable language of finite-order arithmetic, which allows us to quantify over natural numbers, sets of natural numbers, sets of sets of natural numbers, and so on. We measure the complexity of sentences with a generalization of the arithmetical and analytical hierarchies to higher subscripts. We call $\Pi^m_n$ and $\Sigma^m_n$ for $m, n \in \mathbb{N}$ *classes* of the arithmetical hierarchy.

I'm interested in *special* classes of the arithmetical hierarchy that are built up as follows.

 1. $\Delta^0_0$ is special.
 2. If every true sentence in $\Pi^m_n$ (resp., $\Sigma^m_n$) follows from (i.e., can be proved from) true sentences belonging to special classes, then $\Pi^m_n$ (resp., $\Sigma^m_n$) is special.
 3. $\Pi^m_n$ is special if and only if $\Sigma^m_n$ is special.
 4. No other classes are special.

As an example, it's easy to see that $\Pi^0_n$ (equivalently, $\Sigma^0_n$) is special for all $n$. $\Delta^0_0$ is special, and all true $\Sigma^0_1$ sentences follow from true sentences belonging to $\Delta^0_0$ (because the latter contains witnesses for all true $\Sigma^0_1$ sentences). Thus, $\Pi^0_1$ is special. We can then similarly deduce that $\Sigma^0_2$ is special and so on.

If I'm understanding a 1961 result of Grzegorczyk, Mostowski, and Ryll-Nardzewski [1] correctly, then all true $\Pi^1_1$ sentences follow from true first-order arithmetic sentences, so $\Pi^1_1$ is special too.

My question is for which $m$ and $n$ is $\Pi^m_n$ special?

[1] "Definability of sets in models of axiomatic theories" (Thanks to Ali Enayat for bringing this paper to my attention in another question of mine.)