Effective powers of $\omega$ over $\Delta_2$ cohesive sets and infinite $\Pi_1$ sets without $\Delta_2$ cohesive subsets

Abstract

A cohesive power of a computable structure is an effective ultrapower where a cohesive set acts as an ultrafilter. Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of the natural numbers, the integers, and the rationals. We study cohesive powers of computable copies of $\omega$ over $\Delta_2$ cohesive sets. We show that there is a computable copy $\mathcal{L}$ of $\omega$ such that, for every $\Delta_2$ cohesive set $C$, the cohesive power of $\mathcal{L}$ over $C$ has order-type $\omega + \eta$. This improves an earlier result of Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, and Vatev by generalizing from $\Sigma_1$ cohesive sets to $\Delta_2$ cohesive sets and by computing a single copy of $\omega$ that has the desired cohesive power over all $\Delta_2$ cohesive sets. Furthermore, our result is optimal in the sense that $\Delta_2$ cannot be replaced by $\Pi_2$. More generally, we show that if $X \subseteq \mathbb{N} \setminus \{0\}$ is a Boolean combination of $\Sigma_2$ sets, thought of as a set of finite order-types, then there is a computable copy $\mathcal{L}$ of $\omega$ where the cohesive power of $\mathcal{L}$ over any $\Delta_2$ cohesive set has order-type $\omega + \sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$. If $X$ is finite and non-empty, then there is also a computable copy $\mathcal{L}$ of $\omega$ where the cohesive power of $\mathcal{L}$ over any $\Delta_2$ cohesive set has order-type $\omega + \sigma(X)$. An unexpected byproduct of our work is a new method for constructing infinite $\Pi_1$ sets that do not have $\Delta_2$ cohesive subsets. In fact, we construct an infinite $\Pi_1$ set that does not have a $\Delta_2$ p-cohesive subset. Infinite $\Pi_1$ sets without $\Delta_2$ r-cohesive subsets generalize D. Martin's classic co-infinite c.e. set with no maximal superset and have appeared in the work of Lerman, Shore, and Soare.