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Constructive aspects of classical mathematics

古典数学的建设性方面

基本信息

批准号：
0070600
负责人：
Jeremy Avigad
金额：
$ 7.11万
依托单位：
Carnegie-Mellon University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2000
资助国家：
美国
起止时间：
2000-08-01 至 2003-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070600&HistoricalAwards=false
关键词：
Constructive aspects classical mathematics

项目摘要

The investigator is pursuing two lines of research, relevant to thegeneral proof-theoretic program of extracting constructive, computational,and combinatorial information from theories of classical mathematics. Thefirst involves extending a semantic approach to ordinal analysis toKripke-Platek set theory, and extracting combinatorial and computationalprinciples for the constructible hierarchy from the analysis. The secondinvolves trying to find sharp conservation results for weak theories ofnonstandard arithmetic, preferably via "natural" interpretations of thenonstandard theories in the standard ones, and exploring ways in whichordinary mathematics can be carried out in these frameworks.Since the beginning of the 20th century, there has been a gradualdivergence between two different ways of thinking about mathematics. Onthe one hand, mathematics is viewed as a general investigation intoabstract concepts, many of them involving infinitary objects andstructures. On the other hand, many see mathematics as grounded byconcrete symbolic representations and calculation. One goal of prooftheory is to reconcile these two viewpoints, by finding the concrete,computational content that is hidden in general forms of abstractmathematical reasoning. The investigator aims to apply proof-theoreticmethods to the study of theories of nonstandard analysis, and fragments ofset theory.

研究人员正在进行两条线的研究，与从经典数学理论中提取构造性、计算性和组合性信息的一般证明论程序相关。第一种是将序数分析的语义方法扩展到Krigke-Platek集合论，并从分析中提取可构造层次的组合和计算原理。第二种是试图为弱小的非标准算术理论找到明确的守恒结果，最好是通过对标准算术中的非标准理论的“自然”解释，并探索在这些框架中可以进行普通数学的方法。自20世纪初以来，两种不同的数学思维方式之间逐渐出现了分歧。一方面，数学被视为对抽象概念的一般研究，其中许多概念涉及无限对象和结构。另一方面，许多人认为数学是以具体的符号表示和计算为基础的。证明论的一个目标是通过找到隐藏在抽象数学推理的一般形式中的具体的计算内容来调和这两种观点。研究者的目标是将证明论方法应用于非标准分析理论和集合论的碎片研究。