论文标题
可计数集与在反向数学中可计数的集合
Countable sets versus sets that are countable in Reverse Mathematics
论文作者
论文摘要
该程序反向数学（简称RM）旨在识别证明普通数学定理所需的公理，通常以二阶算术$ l_ {2} $的语言工作。因此，RM中的一个主要主题是对可计数的结构的研究，或者可以通过可数集近似。现在，可计数集由这里的序列表示，因为“可计数set'cannot的通常高阶”定义以$ l_ {2} $表示。在Kohlenbach的高阶RM工作，我们研究了各种中央定理，例如那些是由于König，Ramsey，Bolzano，Weierstrass和Borel所致的（通常是原始的）表述，涉及“可计数集”而不是“序列”的通常定义。事实证明，这项研究与作者和DAG Normann最近开发的$ \ Mathbb {r} $的逻辑属性密切相关。现在，可以通过注射到$ \ Mathbb {n} $（kunen）或存在于$ \ Mathbb {n} $（hrbacek-jech）的两者来表示“可计数”。前者（而不是后者）选择会产生“爆炸性”定理，即与不连续功能相结合时相对较弱的陈述，甚至最高$π_2^1 $ -ca $ _0 $。尽管如此，无论使用什么“集合”的概念，用“可计数集”代替“序列”会严重降低这些定理的一阶强度。最后，我们获得了涉及例如König的引理和RM动物园的定理表明，当用可数集配制时，后者是“更驯服的”。
The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic $L_{2}$. A major theme in RM is therefore the study of structures that are countable or can be approximated by countable sets. Now, countable sets are represented by sequences here, because the usual higher-order definition of `countable set'cannot be expressed in $L_{2}$. Working in Kohlenbach's higher-order RM, we investigate various central theorems, e.g. those due to König, Ramsey, Bolzano, Weierstrass, and Borel, in their (often original) formulation involving the usual definition(s) of `countable set' instead of `sequence'. This study turns out to be closely related to the logical properties of the uncountably of $\mathbb{R}$, recently developed by the author and Dag Normann. Now, `being countable' can be expressed by the existence of an injection to $\mathbb{N}$ (Kunen) or the existence of a bijection to $\mathbb{N}$ (Hrbacek-Jech). The former (and not the latter) choice yields `explosive' theorems, i.e. relatively weak statements that become much stronger when combined with discontinuous functionals, even up to $Π_2^1$-CA$_0$. Nonetheless, replacing `sequence' by `countable set' seriously reduces the first-order strength of these theorems, whatever the notion of `set' used. Finally, we obtain `splittings' involving e.g. lemmas by König and theorems from the RM zoo, showing that the latter are `a lot more tame' when formulated with countable sets.
