Countable chains and infinite joins in effectively closed sets of Cantor space

Abstract

We prove that there exists a countable infinite sequence of non-empty special $\Pi^0_1$ classes $\{\mathcal{P}_i\}_{i\in\omega}$ such that no infinite union of elements of any $\mathcal{P}_i$ computes the halting set. We then give a generalized form of lower and upper cone avoidance for infinite unions. That is, we show that for any special $\Pi^0_1$ class $\mathcal{P}$ and any countable sequence of sets in $\mathcal{P}$, $\mathcal{P}$ has a member that is not computable by the infinite union of elements of the sequence. We also prove the upper cone counterpart, that for any non-recursive set $X$, every non-empty $\Pi^0_1$ class contains a countable sequence of members whose join does not compute $X$. We finally show that there exists a $\Pi^0_1$ class whose degree specrum is a countably infinite strict chain.