论文标题
关于实际数字的不可实数的计算属性
On the computational properties of the uncountability of the real numbers
论文作者
论文摘要
实际数字的不可数量是其最基本的属性之一，在数学之外已知（FAR）。 Cantor的1874年证明了实际数字的不可估量，甚至出现在有关集合理论的第一篇论文中，即历史里程碑。尽管存在这种著名的地位和历史，但实际数字不可实数的计算属性并未得到太多研究。在本文中，我们研究了以下计算操作，这些计算操作证明了实数不可数： 在输入一组可数的真实集中，请输出一个真实，而不是在该集合中。 特别是，我们根据基于S1-S9计算方案在Kleene的高阶计算性理论中工作，在计算上等同于中心操作，这在计算上等同于中心操作。也许令人惊讶的是，我们的等效操作涉及Riemann Integrallable和Volterra早期工作的大多数基本特性。
The uncountability of the real numbers is one of their most basic properties, known (far) outside of mathematics. Cantor's 1874 proof of the uncountability of the real numbers even appears in the very first paper on set theory, i.e. a historical milestone. Despite this famous status and history, the computational properties of the uncountability of the real numbers have not been studied much. In this paper, we study the following computational operation that witnesses that the real numbers not countable: on input a countable set of reals, output a real not in that set. In particular, we formulate a considerable number of operations that are computationally equivalent to the centred operation, working in Kleene's higher-order computability theory based on his S1-S9 computation schemes. Perhaps surprisingly, our equivalent operations involve most basic properties of the Riemann integral and Volterra's early work circa 1881.
