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

TL;DR

This paper investigates the structure of cohesive powers of computable copies of  over  cohesive sets, generalizing previous results and introducing new methods for constructing   sets without  cohesive subsets.

Contribution

It extends the understanding of cohesive powers over  cohesive sets, providing a computable copy of  with specific order-types and developing a new method for constructing   sets without  cohesive subsets.

Findings

Existence of a computable   with cohesive powers of order-type  +   over all  cohesive sets.

Generalization of results from   to   sets defined by Boolean combinations of  sets.

Development of a new construction method for infinite   sets lacking   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.