论文标题
关于基本数学概念的计算属性
On the computational properties of basic mathematical notions
论文作者
论文摘要
我们研究了与$ \ mathbb {r} \ rightarrow \ mathbb {r} $ - 函数和$ \ mathbb {r {r} $的子集有关的基本数学概念的计算属性，例如有限性，可计数性，（绝对）连续性，界面变量，跨性变量，跨性别，和常规性。我们基于Kleene的S1-S9方案的高阶可计算理论。我们表明，上述斜体属性产生了两种巨大而强大的计算等效操作类别，后者基于主流数学文献的众所周知的定理。作为这项工作的一部分，我们开发了适合部分对象的S1-S9的等效$λ$ -Calculus公式。我们表明，通过研究$ 3 $的类型$ 3 $的研究对我们的企业至关重要。
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 $λ$-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 $3$.
