What is the Turing degree of the set of true formula of Second Order Arithmetic? The set of true formula of First Order Arithmetic is not arithmetical (by Tarski's undefinability theorem) and it has Turing degree $\emptyset^{(\omega)}$.
What about the set of true formula of Second Order Arithmetic?
I assume we can apply Tarski's undefinability theorem again and saying that this the set is not analytical.
What is the Turing degree of this set?
More in general what is the Turing degree of the set of true formula of $n$-order Arithmetic?
Is there a rule which defines a function which maps the order of the Arithmetic to the ordinal of the Turing degree of the corresponding set of the true formula of that Arithmetic?
 A: I'll focus only on the second-order situation here, since my answer applies a fortiori to the higher orders.
It essentially$^1$ requires us to introduce a new notation, to the point that - in my opinion - true second-order arithmetic (which I'll call "$TA_2$") is fundamentally impossible to describe in a concrete way.
I think it will help to start by considering a simpler example, namely the set of true $\Pi^1_1$ sentences of arithmetic. Up to Turing degree this is just the set of indices of computable well-orderings. We know how to make sense of ${\bf 0^{(\alpha)}}$ for a computable ordinal $\alpha$, but this only gets us hyperarithmetic sets. So we introduce a new idea, the hyperjump, which solves the problem tautologically: this is the map $X\mapsto$ $\mathcal{O}^X$. The $\Pi^1_1$ theory of arithmetic is then just $\mathcal{O}^\emptyset$, or $\mathcal{O}$.
(I'm being a bit sloppy here, and in particular ignoring the distinction between computable well-orderings and notations; this is a bit technical and doesn't affect the Turing degree involved.)
One level higher, we hit the set of true $\Pi^1_2$ sentences (or thinking structurally, the set of indices for computable examples of a more complicated type of object called a dilator). One might reasonably expect (in analogy with the arithmetic case, and the usual Turing jump) that this is just $\mathcal{O}^\mathcal{O}$. However, this is much, much, much, ... , much more complicated than anything you can get by iterating the hyperjump a "reasonable" number of times, in the same way that $\mathcal{O}$ itself is more complicated than anything you can get by applying the usual Turing jump a "reasonable" number of times (the latter being the hyperarithmetic sets).
And this happens at each subsequent level: the "$\Pi^1_k$-jump" is liliputian compared to the "$\Pi^1_{k+1}$-jump." One would really have to introduce new symbols for each of these. (Incidentally, on the "structural" side the generalization of ordinals and dilators is ptykes, introduced by Girard).
Of course, this isn't really the end of the story. Even granting the above we can still try to understand $TA_2$ by comparing it to structures/sets we find more concrete. For example, the first-order theory of the partial order of Turing degrees (and some other degree structures) is $1$-equivalent to $TA_2$. Meanwhile, many of the important reals we get from large cardinals are strictly less complicated than $TA_2$ if they exist in the first place (e.g. $0^{\sharp}$ is $\Delta^1_3$). But I think these observations really go the other way: e.g. "$Th_{FOL}(\mathcal{D})$ computes $TA_2$" doesn't concretify $TA_2$, it reveals that $\mathcal{D}$ is really mysterious.

$^1$That's not quite true, but I'd say it's still pretty morally correct. Here are two amendations: 


*

*If $V=L$, we can talk about iterating the Turing jump past $\omega_1^{CK}$ via mastercodes. Roughly speaking, in the mastercode hierarchy the true theory of second-order arithmetic will appear at level $\alpha$, where $\alpha$ is the lowest level of the $L$-hierarchy whose set of reals satisfies that theory. But this only works assuming $V=L$ (or similar) and is also fairly tautological; moreover, the mastercode hierarchy has some annoying pathologies. So while interesting, this isn't really satisfying.

*At the other end of the set-theoretic spectrum and considering first a "higher-type" analogue of this problem, it turns out that reasonably-strong large cardinal hypotheses imply that "natural" operations on Turing degrees are (pre-)well-ordered in a particular way. The relevant term here is "Martin's Conjecture" (MC). While the full MC is still open, much partial progress is known, and in particular Steel's proof of one half of its weaker version the relevant part of MC for uniformly representable functions is applicable to our situation: the relevant function $$TSOA: r\mapsto Th_{SOL}(\mathbb{N},\mathcal{P}(\mathbb{N}),+,\cdot,\in,r)$$ lives in a particularly natural pre-well-ordering. It therefore has a naturally-associated ordinal, namely its rank $\sigma$ in this pre-well-ordering. Getting back to the original question, one might hope that "the second-order theory of $\mathbb{N}$ has degree ${\bf 0^{(\sigma)}}$" is true in some sense, but I actually don't know a precise interpretation of this. Ultimately I think that this kind of analysis-by-ordinals is not really addressing the concrete question of the OP although it is quite interesting.
