Tennenbaum-like theorems for cohesive powers

David Gonzalez; Paul Shafer

Tennenbaum-like theorems for cohesive powers

Logic 2026-08-05 v1

Authors: David Gonzalez , Paul Shafer

Abstract

We investigate the encoding ability of the cohesive power construction. We compute a graph $\mathcal{G}$ where the cohesive power $\prod_C \mathcal{G}$ of $\mathcal{G}$ by any $\Delta_2$ cohesive set $C$ has degree $0''$ . That is, $0''$ computes a presentation of $\prod_C \mathcal{G}$ , and every presentation of $\prod_C \mathcal{G}$ computes $0''$ . We also compute a linear order $\mathcal{L}$ where no cohesive power of $\mathcal{L}$ has a computable presentation. We accomplish this by ensuring that if $\mathcal{P}$ is a presentation of a cohesive power of $\mathcal{L}$ , then $\mathcal{P}''$ has $\mathrm{PA}$ -degree relative to $0''$ .

Cite

@article{arxiv.2608.04654,
  title  = {Tennenbaum-like theorems for cohesive powers},
  author = {David Gonzalez and Paul Shafer},
  journal= {arXiv preprint arXiv:2608.04654},
  year   = {2026}
}