8 problems

Slaman–Woodin conjecture. The set

Let $A$ be a computably enumerable set whose complement is semilow. Effective automorphism conjecture. $A$ is effectively automorphic to a low set. Soare proved that the lattice of…

Let $A$ be a c.e. set. It has the outer splitting property if there are computable functions $f,h$ such that, for every $e$, $W_e=W_{f(e)}\sqcup W_{h(e)}$,…

Let $A$ be a c.e. set. Its complement $\overline{A}$ is semilow when the corresponding semilowness property holds, and $A$ is low when $A'\equiv_T\mathbf{0}'$. Two c.e. sets are au…

For a c.e. set $A$, let $\mathcal{L}(A)$ be the collection of c.e. supersets of $A$ under inclusion, let $\mathcal{F}$ be the filter of finite sets, and let…

Let $A$ be a c.e. set whose orbit is properly $Delta^0_{\alpha}$ for a finite $\alpha>8$, as in Theorem 3.2, and let $\mathcal{L}_{\omega_1,\omega}$ denote the infinitary logic use…

Nonlow-degree conjecture. C.e. sets which are generically computable but not coarsely computable exist only in nonlow c.e. degrees.

Let $A$ be a computably enumerable set and let $\boldsymbol{d}$ be a Turing degree. Write $A'$ for the Turing jump of $A$, and let $\mathcal{L}^*(A)$ denote the lattice structure a…