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

TL;DR

This paper explores the structure of effectively closed sets in Cantor space, demonstrating the existence of countable chains and infinite joins with specific computability properties, advancing understanding in computability theory.

Contribution

It establishes the existence of countable chains and infinite joins in effectively closed sets with particular non-computability characteristics, generalizing cone avoidance results.

Findings

Existence of a countable infinite sequence of special $\

Construction of a $\

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.