Introenumerability, autoreducibility, and randomness

Abstract

We define $\hat{I}¨$-autoreducible sets given an autoreduction procedure $\hat{I}¨$. Then, we show that for any $\hat{I}¨$, a measurable class of $\hat{I}¨$-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly introenumerable, introenumerable, and hyper-cototal enumeration degrees all have measure zero. By analyzing the arithmetical complexity of the classes of cototal sets and cototal enumeration degrees, we show that weakly 2-random sets cannot be cototal and weakly 3-random sets cannot be of cototal enumeration degree. Then, we see that this result is optimal by showing that there exists a 1-random cototal set and a 2-random set of cototal enumeration degree. For uniformly introenumerable degrees and introenumerable degrees, we utilize $\hat{I}¨$-autoreducibility again to show the optimal result that no weakly 3-random sets can have introenumerable enumeration degree. We also show that no 1-random set can be introenumerable.