Reverse Math of High Sets? Is there a standard principle in reverse math that is known to be equivalent (over $RCA_0$) to the existence of a set of high (Turing) degree? I'm interested in the general case, but would be happy to learn of such a principle for $\omega$-models.
I haven't been able to find much discussion on this topic... but then, I don't have much experience in reverse math yet. If the answer is obvious (say, $ACA_0$), please forgive me!
To clarify: the specific principle I'm interested in is the statement that "for all $X$, there exists some $Y\ge_T X$ with $Y'=X''$", appropriately rephrased to avoid the explicit use of the jump operator.
 A: In the paper "On a conjecture of Dobrinen and Simpson regarding almost everywhere domination", Binns, Lerman, Solomon and I constructed $\omega$-models of this "high" principle which demonstrate it does not imply WKL, WWKL, or $G_\delta$-regularity.
A: I believe the answer is "no." Certainly $ACA_0$ is overkill; we can build an ascending sequence $H_0\le_T H_1\le_T . . .$ of sets such that $H_{n+1}'=H_n''$ (i.e., $H_{n+1}$ is high over $H_n$), but no $H_n$ computes $0'$. This is done by iterating a jump inversion theorem:

Suppose $X$, $Y$, and $Z$ are sets such that $X=Z'$ and $Z\not\ge_T Y$. Then there is a set $A$ with $A'=X$, $A\ge_T Z$, and $A\not\ge_T Y$.

(If I recall correctly, this is basically the Friedberg Completeness Criterion with extra bells and whistles, although there are several jump inversion theorems, so I might have the wrong name.) To get $H_{n+1}$ from $H_n$, just apply the theorem with $X=H_n'$, $Z=H_n$, and $Y=0'$.
On the other hand, such a principle might be useful in reverse math down the road, even if it hasn't shown up yet. A while ago, Damir Dzhafarov and I thought we'd found a use for the principle "for each set $X$, there is a $Y\ge_TX$ with $Y'=X''$" (rephrased appropriately, since of course jumps need not exist). Now if I recall correctly it wound up not being strong enough to help us, but I still think it's probably relevant for something. 
