Measuring the Complexity of Countable Presburger Models
Jason Block
TL;DR
The paper studies the descriptive and computability-theoretic complexity of countable Presburger models by combining Scott analysis with degree spectra. It constructs Presburger groups $P_{\\mathcal{L}}=V_{\\mathcal{L}}\times\\mathbb{Z}$ from countable linear orders $\\mathcal{L}$ by taking $V_{\\mathcal{L}}=\bigoplus_{l\in\\mathcal{L}}\\mathbb{Q}$ to preserve much of the order-theoretic structure, enabling precise control of infinitary descriptions and copies. It proves that Presburger groups realize a wide array of Scott sentence complexities, including $\\Pi_{\\alpha}^{in}$, $d$-$\\Sigma_{\\alpha}^{in}$ for successor $\\alpha$, and $\\Sigma_{\\alpha}^{in}$ for certain successors, while ruling out $\\Sigma_{3}^{in}$, and shows that Presburger arithmetic has a complete Scott spectrum. On the degree-spectrum side, it shows $DgSp(P_{\\mathcal{L}})=\{\\boldsymbol{d}:\\boldsymbol{d}'\in DgSp(\\mathcal{L})\}$ and develops constructions that realize spectra of the form $\{\\boldsymbol{d}: S\text{ is computable in }\\boldsymbol{d}^{(\\alpha)}\}$ or $\{\\boldsymbol{d}: S\text{ is c.e. in }\\boldsymbol{d}'\}$, with specialized results for one-jump spectra and for spectra arising from the degree structure of the underlying linear order. The work also discusses non-plain and recursively saturated Presburger groups, establishing their Scott rank 2 and highlighting open questions about the full landscape of Scott complexities and degree spectra in this setting. These results illuminate the richness of Presburger models, offering a bridge between linear-order complexity and Presburger-group complexity and indicating both the limits and potentials of degree-spectrum universality in this domain.
Abstract
We take two approaches to classifying the complexity of Presburger models: Scott analysis and degree spectra. In particular, we investigate the possible Scott sentence complexities and possible degree spectra of models of Presburger arithmetic. Many of our results will be achieved by showing how given a linear order $\mathcal{L}$, we can construct a Presburger group $P_\mathcal{L}$ that maintains much of the structure of $\mathcal{L}$.