Countability constraints in order-theoretic approaches to computability

TL;DR

This paper explores how countability constraints in order-theoretic models influence the formalization of computability on uncountable sets, linking these constraints to order density and utility representations.

Contribution

It establishes new relations between classical countability restrictions and order-theoretic properties, enabling the introduction of computability in certain order structures.

Findings

Countability restrictions relate to order density properties.

Multi-utility representations characterize order structures.

Computability can be formalized using these order-theoretic constraints.

Abstract

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.