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.