In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with deciding extensional equality. Using homotopy type theory as the foundational setting, we develop an abstract framework for ordinal theory and establish a collection of desirable properties and constructions. We then study and compare three concrete implementations of ordinals in homotopy type theory: first, a notation system based on Cantor normal forms (binary trees); second, a refined version of Brouwer trees (infinitely-branching trees); and third, extensional well-founded orders. Each of our three formulations has the central properties expected of ordinals, such as being equipped with an extensional and well-founded ordering as well as allowing basic arithmetic operations, but they differ with respect to what they make possible in addition. For example, for finite collections of ordinals, Cantor normal forms have decidable properties, but suprema of infinite collections cannot be computed. In contrast, extensional well-founded orders work well with infinite collections, but almost all properties are undecidable. Brouwer trees take the sweet spot in the middle by combining a restricted form of decidability with the ability to work with infinite increasing sequences. Our three approaches are connected by canonical order-preserving functions from the "more decidable" to the "less decidable" notions. We have formalised the results on Cantor normal forms and Brouwer trees in cubical Agda, while extensional well-founded orders have been studied and formalised thoroughly by Escardo and his collaborators. Finally, we compare the computational efficiency of our implementations with the results reported by Berger.
翻译:在建设性环境下,没有具体方程式可以同时具有人们可能感兴趣的所有属性; 例如, 能够计算序列的界限与决定扩展平等不相容。 我们使用单式型理论作为基础设置, 开发一个用于正态理论的抽象框架, 并建立一个适合的属性和构造的集合。 然后我们研究和比较在同质类型理论中, 方言的三种具体实施方式: 首先, 基于 Cantor 常规形式的标记系统( 双层树 ); 第二, 精细版的布鲁威尔树( 无限布林树 ) ; 第三, 扩展型树木的顺序与决定扩展型平等不相容。 我们的三种配方言词, 配有扩展型和完善的排序, 并允许基本的算术操作操作, 但是, 与它们之间有不同的是。 例如, 坎托尔的常规形式有易变异的特性, 但是不易变现的正规形式是变的。 对比的是, 扩展型的直立型树木的顺序与不易变的顺序, 最终将我们的工作顺序与不易变的顺序结合, 。 最后, 以平质的顺序 以平质的顺序研究, 将我们的工作顺序 以平质的顺序, 以平质的顺序 以平的顺序 以平的顺序 以平的顺序 以平的顺序 以平的顺序 以平的 以平的 以平的顺序 以平的顺序 的顺序 以平的 以平的 以平的 以平态形式报告 的 以平式形式报告 。