A classic result of Jockusch states that, for every k >= 2, every
DNR(k) function computes a DNR(2) function. Jockusch's proof can be
formalized in RCA_0 + ISigma_2, thus the corresponding reverse
mathematics result is that the principles "exists k DNR(k)" (i.e.,
there is a k such that for every X there is a function that is DNR(k)
relative to X) and WKL are equivalent over RCA_0 + ISigma_2. The
question then, perhaps first articulated by Simpson, is whether or not
ISigma_2 is necessary. In a step toward separating the principles
"exists k DNR(k)" and WKL over RCA_0, we show that if BSigma_2 holds
and ISigma_2 fails, then there is a DNR(k) function (for some
necessarily non-standard k) that computes no DNR(2) function. This is
joint work with Francois Dorais (Dartmouth College) and Jeffry Hirst
(Appalachian State University).