  In a previous paper, the author introduced the idea of intrinsic density ---
a restriction of asymptotic density to sets whose density is invariant under
computable permutation. We prove that sets with well-defined intrinsic density
(and particularly intrinsic density 0) exist only in Turing degrees that are
either high ($\mathbf{a}'\ge_{\rm T}\emptyset"$) or compute a diagonally
non-computable function. By contrast, a classic construction of an immune set
in every non-computable degree actually yields a set with intrinsic lower
density 0 in every non-computable degree.
  We also show that the former result holds in the sense of reverse
mathematics, in that (over $\mathsf{RCA}_0$) the existence of a dominating or
diagonally non-computable function is equivalent to the existence of a set with
intrinsic density 0.
