2025/11/06 by J. Block, Block, Jason, Russell Miller +1
Computer Science · Mathematics · #03C57 (Secondary) #03D45 (Primary) 03D78 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2511.04152
openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28
We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the focus to first-order properties, using computable structure theory (on countable structures) as a guide. We present three basic questions to be asked when a structure is presented as the set of paths through a computable tree, as in our definition, and also propose the concept of tree-decidability as an analogue to the notion of decidability for a countable structure. As examples, we prove decidability results for certain additive and multiplicative groups of p-adic integers, products of these (such as the profinite completion of \mathbb Z), and the field of real numbers.