论文标题
$ \ mathbb {r} $的不可实数的反向数学
Reverse Mathematics of the uncountability of $\mathbb{R}$
论文作者
论文摘要
Cantor在他的第一盘理论论文（1874年）中确立了$ \ Mathbb {r} $的不可估量。我们在Kohlenbach的高阶反向数学中研究了后者，这是由观察到的观察，即无法研究诸如“从$ \ Mathbb {r} $到$ \ Mathbb {n} $”的概念。现在，最近显示出以下语句：$$ \ text {nin：从$ [0,1] $到$ \ mathbb {n} $，} $ 4的注入很难在常规理解方面证明。在本文中，我们表明NIN通过建立NIN和NIN之间的等价限制为主流功能类，例如：有界变化，半连续性和Borel。因此，NIN的上述硬度并不是由于对任意$ \ Mathbb {r} \ rightArrow \ Mathbb {n} $ - 函数NIN的量化的量化。最后，我们还研究了NBI，NIN限制对射对，以及与堂兄的引理和约旦分解定理的联系。
In his first set theory paper (1874), Cantor establishes the uncountability of $\mathbb{R}$. We study the latter in Kohlenbach's higher-order Reverse Mathematics, motivated by the observation that one cannot study concepts like `arbitrary mappings from $\mathbb{R}$ to $\mathbb{N}$' in second-order Reverse Mathematics. Now, it was recently shown that the following statement: $$ \text{ NIN: there is no injection from $[0,1]$ to $\mathbb{N}$,} $$ is hard to prove in terms of conventional comprehension. In this paper, we show that NIN is robust by establishing equivalences between NIN and NIN restricted to mainstream function classes, like: bounded variation, semi-continuity, and Borel. Thus, the aforementioned hardness of NIN is not due to the quantification over arbitrary $\mathbb{R}\rightarrow \mathbb{N}$-functions in NIN. Finally, we also study NBI, the restriction of NIN to bijections, and the connection to Cousin's lemma and Jordan's decomposition theorem.
