What is the Turing degree of truth in the second-order theory of real numbers? Let $X$ be the set of Godel numbers of sentences in the second-order language of ordered fields which are true in $\mathbb{R}$.  Then my question is, what is the Turing degree of $X$?
In particular, how does it compare to the Turing degree of truth in second-order arithmetic?  One datapoint is that the second-order arithmetic is interpretable in the second-order theory of real numbers.  Another datapoint is second-order theory of real numbers is interpretable in second-order arithmetic with third-order parameters.  Yet another datapoint is that both second-order arithmetic and the second-order theory of real numbers are categorical theories.  I'm not sure whether these facts imply that the Turing degree of $X$ is the same as the Turing degree of truth in second-order arithmetic, but it least seems intuitively plausible that the two Turing degrees are of the same "order of magnitude".
 A: Since $\mathbb{N}$ is second-order-definable in $\mathbb{R}$, the true second-order theory of $\mathbb{R}$ is basically the same as (= computably isomorphic to) true third-order arithmetic. And the gap between true third-order arithmetic and true second-order arithmetic is gigantic. For example, true second-order arithmetic (construed as a set of natural numbers via Godelization) is definable over $\mathbb{N}$ by a single third-order formula.
Incidentally, there's a methodological point here: what would a "satisfying description" of the relevant Turing degree consist of? For sets of this complexity we're way beyond the point of well-behaved degree constructions, and I think the best we can hope for is descriptions of the form "The Turing degree of the $\mathfrak{L}$-theory of $\mathcal{M}$" for some natural logic $\mathfrak{L}$ and natural structure $\mathcal{M}$. It's often convenient to look for descriptions of this form with $\mathcal{M}$ a nice transitive set, so in this case "The third-order theory of $L_\omega$" would be my suggestion of an answer.
(This point about "naming" Turing degrees is similar to the point about "naming" large ordinals I made here, and I believe in a couple other related answers as well. In each case, unless one has a specific form of description in mind, I think the initial description is actually optimal-ish.)
