Status of Open Questions

Below are some resolutions to questions I have asked, along with links/attributions where necessary. Last Updated 9/13/2020

Intrinsic Smallness, 2020:

  • Previously question 4.2: There is an arithmetically intrinsically small set which is not omega-th jump intrinsically small. In fact, there is an omega-th jump computable such set, as the omega-th jump can uniformly compute the n+2nd jump, and therefore can uniformly compute all arithmetical permutations. (This has been included in the most recent version of the paper.)
  • Question 4.1: For all X, the X-degrees are the X-high or X-DNC degrees. The proof appears in subsection 3.1 of Noncomputable Coding, Density, and Stochasticity.