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.)



Special classes of the arithmetical hierarchy of sentences of finite-order arithmetic
Per the comments, we're looking at deduction in some system based on the $\omega$-rule as opposed to standard first-order deduction (or Henkin semantics or etc.). There's a technical issue here - in my experiene the $\omega$-rule is usually formulated for *first-order arithmetic* sentences, so I'm not sure what it means to deduce a $\Pi^m\_n$ or $\Sigma^m\_n$ sentence using the $\omega$-rule for $m>0$ (this may be in the linked paper I don't have access to) - but in fact there's a coarse calculation which will apply to any reasonable interpretation I can think of:


Every version of the $\omega$-rule I can think of is $\Pi^1\_1$ - roughly, "the $\omega$-consequences of $T$" will always be $\Pi^1\_1$ relative to $T$. If (for example) we start with the true $\Pi^1\_1$ theory of arithmetic, we won't even get all the true $\Sigma^1\_1$ sentences since the true $\Sigma^1\_1$ theory of arithmetic isn't itself $\Pi^1\_1$ (more broadly, the projective hierarchy doesn't collapse).


Now your notion of specialness gives us only two tools for "climbing up" the syntactic hierarchy: deduction and complementation. In terms of Turing degree this means that we're not going to escape the $\omega$th hyperjump of $\emptyset$, which is a tiny subclass of $\Delta^1\_2$.


*(This is contra a silly claim I made originally - the point is that "$\Pi^1\_1$ in $\Sigma^1\_1$" is much *weaker* than "$\Pi^1\_2$," or more concretely that $\mathcal{O}^\mathcal{O}$ is not $\Pi^1\_2$ complete. The sense in which applying the $\omega$-rule "adds a $\Pi^1\_1$" seems to follow the former pattern if set up in a natural way. That said, I see no sense at all in which we can get outside the analytical hierarchy.)*


