Representations and the Foundations of Mathematics

IF 0.6 3区 数学 Q2 LOGIC
Sam Sanders
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引用次数: 3

Abstract

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.
表征与数学基础
用(更)基本的数学对象来表示数学对象是数学基础的一部分。在通常的数学基础中,即$\textsf{ZFC}$集合论,所有数学对象都用集合表示,而普通的,即非集合论,数学用二阶算术的更简约的语言表示。本文讨论了实空间和Baire空间上连续函数的基本情形的后一种表示。我们证明了以Tietze、Heine和Weierstrass命名的基本定理的逻辑强度在“二阶表示”替换为“三阶函数”时发生了显著变化。我们讨论了逆向数学程序的含义和联系,以及它关于谓词数学和希尔伯特数学基础程序的基本主张。最后,我们确定了由连续函数的表示引起的问题,并制定了一个标准,以避免在表示的更大范围内进行有问题的编码。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.00
自引率
14.30%
发文量
14
审稿时长
>12 weeks
期刊介绍: The Notre Dame Journal of Formal Logic, founded in 1960, aims to publish high quality and original research papers in philosophical logic, mathematical logic, and related areas, including papers of compelling historical interest. The Journal is also willing to selectively publish expository articles on important current topics of interest as well as book reviews.
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