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.



Reverse Math of High Sets?
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.


