This question was originally posted as a part of this other question, but I was suggested to make a new question for this part.
In the first question I asked about the Turing degree of the set of true formula of $n$-order Arithmetic, where $n$ is a finite ordinal. While I am writing the first question has not received any answer yet.
I assume you can define orders of Arithmetic corresponding to infinite ordinals because following the definition of Beth numbers I can use transfinite recursion on the power set operator: $$S_0 = \mathbb{N}$$ $$S_{\alpha+1} = P(S_{\alpha})$$
where $P$ is the power set operator and for limit ordinals $$ S_\lambda = \bigcup _{\beta <\lambda }S_{\beta }$$
then $\text{card}(S_\alpha) = \beth_\alpha$
So I would like to ask what is the Turing degree of the set of true formula of $\alpha$-order Arithmetic, where $\alpha$ is an infinite ordinal.
I would like just to end this question with some remarks, I don't see much focus on Arithmetic above the second order. What is the reason?
I try to make a guess. For the purpose of reverse mathematics, most of common mathematics (I mean the mathematics done by mathematician who are not concerned with foundations) is almost entirely contained by Second Order Arithmetic. And for the things which are beyond that you can use ZFC.
A second reason may be that reverse mathematics is a relatively young field (few decades old) with not so many people involved and so there has not yet been enough time to develop results for higher order arithmetics. This is my guess assuming that only people in reverse mathematics may be interested in higher order arithemtics. I am very curios to hear about other explanations.