编码不具有稳健性
Coding is non-robust
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中文总结 AI 辅助
本文研究逆数学中开集编码的稳健性，证明编码的轻微变化会使开集基本性质蕴含强系统ATR0，并给出与枚举原理的等价性，同时探讨闭包、内部和边界的RM性质。
中文摘要 AI 辅助
由于各种原因，高阶对象在数理逻辑中常常通过二阶“编码”或“表示”来研究。一个源自实分析和拓扑学的重要例子是开集，它由并集$\cup_{n\in\mathbb{N}}I_{n}$表示，其中每个$I_{n}$是基本开区间。如今，Montalbán最近强调了逆数学（简称RM）中逻辑系统稳健性的重要性。因此，一个自然的RM问题是：开集的基本性质在开集编码的轻微修改下是否稳健。在这里，我们研究这样的表示：上述$I_{n}$要么是开区间，要么是两个这样的区间的并集，但我们无法决定是哪一种。在这种对通常编码的轻微变化下，开集和闭集的基本性质立即蕴含了RM中相对较强的系统ATR$_{0}$。此外，我们获得了前述性质与“枚举原理”之间的等价性。后者断言可数集可以被枚举，并且在傅里叶分析中拥有许多等价形式。在此过程中，我们研究了实数集合的闭包、内部和边界在RM中的性质，这一研究本身也颇具趣味。
英文摘要
For various reasons, higher-order objects are often studied in mathematical logic via second-order `codes' or `representations'. An important example hailing from real analysis and topology is provided by open sets, which are represented by unions $\cup_{n\in \mathbb{N}}I_{n}$ where each $I_{n}$ is a basic open interval. Now, Montalbán has recently highlighted the importance of robustness of logical systems in Reverse Mathematics (abbreviated RM). It is then a natural RM-question whether basic properties of open sets are robust under slight modifications of the coding of open sets. Here, we study the representation where $I_{n}$ as above is \emph{either} an open interval \emph{or} the union of two such intervals \emph{but} we cannot decide which one. Under this slight variation of the usual coding, basic properties of open and closed sets readily imply the relatively strong system ATR$_{0}$ from RM. Moreover, we obtain equivalences for the former properties and the \emph{enumeration principle}. The latter states that countable sets can be enumerated and boasts many equivalences from Fourier analysis. Along the way, we investigate the RM-properties of the closure, interior, and boundary of sets of reals, a study interesting in its own right.
发表机构
- Department of Philosophy II, RUB Bochum（鲁尔大学波鸿分校哲学第二系）
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