Status of Some Open Questions

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

Last Updated: 7/29/2021

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 arithmetical permutations.
• Question 4.1: For all \(X\), the degrees of \(X\)-intrinsically small sets are exactly the \(X\)-high or \(X\)-DNC degrees. (This is Corollary 2.27 in my thesis.)

Dissertation

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