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Extensional constructive real analysis via locators
Mathematical Structures in Computer Science ( IF 0.5 ) Pub Date : 2020-09-02 , DOI: 10.1017/s0960129520000171
Auke B. Booij

Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known. We overcome this by considering real numbers equipped with additional structure, which we call a locator. With this structure, it is possible, for instance, to construct a signed-digit representation or a Cauchy sequence, and conversely, these intensional representations give rise to a locator. Although the constructions are reminiscent of computable analysis, instead of working with a notion of computability, we simply work constructively to extract observable information, and instead of working with representations, we consider a certain locatedness structure on real numbers.

中文翻译:

通过定位器进行扩展构造实分析

实数不允许观察离散信息的扩展过程,例如其十进制扩展的第一个数字,因为众所周知,从实数到整数的每个可扩展的可计算映射都是恒定的。我们通过考虑配备额外结构的实数来克服这个问题,我们称之为定位器。使用这种结构,例如,可以构建有符号数字表示或柯西序列,相反,这些内涵表示会产生定位符。尽管这些结构让人想起可计算分析,但我们没有使用可计算性的概念,而是简单地建设性地提取可观察信息,而不是使用表示,而是考虑实数上的某种定位结构。
更新日期:2020-09-02
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