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A cohesive power of a computable structure is an effective ultrapower where acohesive set acts as an ultrafilter. Let $\omega$, $\zeta$, and $\eta$ denotethe respective order-types of the natural numbers, the integers, and therationals. 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 cohesivepower of $\mathcal{L}$ over $C$ has order-type $\omega + \eta$. This improvesan earlier result of Dimitrov, Harizanov, Morozov, Shafer, A. Soskova, andVatev by generalizing from $\Sigma_1$ cohesive sets to $\Delta_2$ cohesive setsand by computing a single copy of $\omega$ that has the desired cohesive powerover all $\Delta_2$ cohesive sets. Furthermore, our result is optimal in thesense that $\Delta_2$ cannot be replaced by $\Pi_2$. More generally, we showthat 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 acomputable 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 andnon-empty, then there is also a computable copy $\mathcal{L}$ of $\omega$ wherethe cohesive power of $\mathcal{L}$ over any $\Delta_2$ cohesive set hasorder-type $\omega + \sigma(X)$. An unexpected byproduct of our work is a newmethod 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 nothave 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 nomaximal superset and have appeared in the work of Lerman, Shore, and Soare.

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