We investigate the computational properties of basic mathematical notions pertaining to $\mathbb{R}\rightarrow \mathbb{R}$-functions and subsets of $\mathbb{R}$, like finiteness, countability, (absolute) continuity, bounded variation, suprema, and regularity. We work in higher-order computability theory based on Kleene's S1-S9 schemes. We show that the aforementioned italicised properties give rise to two huge and robust classes of computationally equivalent operations, the latter based on well-known theorems from the mainstream mathematics literature. As part of this endeavour, we develop an equivalent $\lambda$-calculus formulation of S1-S9 that accommodates partial objects. We show that the latter are essential to our enterprise via the study of countably based and partial functionals of type $3$.
翻译:我们根据克莱恩的S1-S9计划,研究与美元有关的基本数学概念的计算特性。 我们发现,上述的斜体化特性产生了两个庞大和强大的计算等效操作类别,后者基于主流数学文献中众所周知的理论。 作为这项努力的一部分,我们开发了一个等效的S1-S9的等值的S1-S9计算公式配方。我们通过对3美元类型的可计算基础和部分功能的研究,表明后者对我们企业至关重要。