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.

翻译：与可数集上的可计算性不同，不可数集上的可计算性缺乏标准的形式化定义，而前者由图灵机给出。在这些集合中定义可计算性的一些方法依赖于序论结构，以将此类概念从图灵机推广到不可数空间。由于这些方法以图灵机作为可计算性的基准，因此序结构的可数性约束具有根本重要性。本文揭示了序论可计算性理论中常见的可数性约束与一些更常见的序论可数性条件之间的关系，例如序稠密性以及通过多效用函数对序结构进行函数刻画。作为结果，我们展示了如何通过可数序稠密性和多效用约束在某些序结构中引入可计算性。