In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. Ulrich Kohlenbach introduced abstract types for certain structures such as metric spaces, normed spaces, Hilbert spaces, etc. With these types, the elements of the spaces are given directly, not through the mediation of a representation. However, these abstract spaces presuppose the real numbers. In this paper, we show how to set up an abstract type for the real numbers. The appropriateness of our construction works in tandem with the bounded functional interpretation.

中文翻译：

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实数的抽象类型

在有限类型算术中，实数由有理数的快速收敛柯西序列表示。乌尔里希·科伦巴赫（Ulrich Kohlenbach）为某些结构（例如度量空间，范数空间，希尔伯特空间等）引入了抽象类型。使用这些类型，空间的元素是直接给出的，而不是通过表示的中介来给出的。但是，这些抽象空间以实数为前提。在本文中，我们展示了如何为实数设置抽象类型。我们的建筑工程的适当性与有限的功能诠释是一脉相承的。