# Paul Shafer (University of Leeds)

- Date
- Wednesday 12 October 2022, 4:00 PM
- Location
- MALL
- Category
- Logic Seminar

## Cohesive powers of linear orders

A cohesive power of a computable structure is an effective analog of an ultrapower, where a cohesive set acts as an ultrafilter. We compare the properties of cohesive powers to those of classical ultrapowers. In particular, we investigate what structures arise as the cohesive power of $B$ over $C$, where $B$ varies over the computable copies of some fixed computably presentable structure $A$, and $C$ varies over the cohesive sets.

Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of $(\mathbb{N}, <)$, $(\mathbb{Z}, <)$, and $(\mathbb{Q}, <)$. We take omega as our computably presentable structure, and we consider the cohesive powers of its computable copies. If L is a computable copy of omega that is computably isomorphic to the standard presentation $(\mathbb{N}, <)$, then all of $L$'s cohesive powers have order-type $\omega + (\zeta \times \eta)$, which is familiar as the order-type of countable non-standard models of PA.

We show that it is possible to realize a variety of order-types other than $\omega + (\zeta \times \eta)$ as cohesive powers of computable copies of omega. For example, we show that there is a computable copy $L$ of omega whose power by any $\Delta_2$ cohesive set has order-type $\omega + \eta$. More generally, we show that it is possible to achieve order-types of the form $\omega +$ certain shuffle sums as cohesive powers of computable linear orders of type $\omega$.