Open Questions

This is a list of solutions to open questions appearing in previous work, along with links and attributions where necessary.

Last Updated: 8/1/2023

Intrinsic Smallness

• Previously Question 4.2: There is an arithmetically intrinsically small set which is not \(\emptyset^{(\omega)}\)-intrinsically small. In fact, there is an \(\emptyset^{(\omega)}\)-computable set with this property, as \(\emptyset^{(\omega)}\) can compute \(\emptyset^{(n+2)}\) uniformly in \(n\) and therefore can compute all arithmetic permutations uniformly.
• Question 4.1: For all \(X\), the degrees of \(X\)-intrinsically small sets are exactly the \(X\)-high or \(X\)-DNC degrees. (Corollary 2.27 in my dissertation)

Dissertation

• Previously Question 4.2.6: The degrees of intrinsic density \(r\) sets cannot be the high or DNC degrees for all \(r\) because there are uncountably many \(r\) but only countably many elements of each degree. This was observed by Denis Hirschfeldt.