Computability for tree presentations of continuum-size structures
Jason Block, Russell Miller
TL;DR
The paper develops a framework to study computability on continuum-sized structures via tree presentations, bridging computable structure theory with continuum topology. It formalizes tree presentations and tree quotients and applies them to $\mathbb{Z}_p^+$, $\mathbb{Z}_p^{\textsf{x}}$, $\widehat{\mathbb{Z}}$, and $\mathbb{R}$ to analyze elementarity, Skolem functions, and definable sets. Key results include: every Turing ideal $I$ yields an elementary subgroup $(\mathbb{Z}_p^+)_I$, properly existential-atomic formulas have computable Skolem functions in $T_p^+$, and $T_p^+$, $T_p^{\textsf{x}}$, and $\widehat{\mathbb{Z}}$ are tree-decidable, while certain infinite products are not. The framework also shows how computable isomorphisms between presentations preserve computable-definable properties and enables transfer of results across continuum-sized contexts.
Abstract
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.