Research in the Foundations of Mathematics

数学基础研究

• 批准号: 0245349
• 负责人: Harvey Friedman
• 金额: $ 21.6万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245349&HistoricalAwards=false
• 关键词: Research; Foundations; Mathematics

Award: DMS-0245349Principal Investigator: Harvey M. FriedmanFriedman proposes to continue his efforts into extending thescope of the incompleteness phenomena. Under prior NSF support,Friedman has discovered a new mathematical theory which seeks toanalyze the Boolean relations that hold between sets and theirimages under functions of several variables. This new Booleanrelation theory seeks to analyze statements of the form "for allfunctions of a certain kind, there exist sets of a certain kind,such that a given Boolean relation holds among the sets and theirimages under the functions". Under prior NSF support, Friedmandiscovered a "singular" statement of a particularly simple formin Boolean relation theory that can be proved only by goingbeyond the usual axioms for mathematics. Friedman has consideredall 6561 statements of the same simple form and showed that allcan be proved or refuted using weak axioms, with the soleexception (up to symmetry) of the "singular" statement. Friedmanproposes to develop Boolean relation theory in severaldirections, including expanding the set of 6561 statements, andshifting to many diverse mathematical contexts.By the early part of the 20th century, the standard axioms andrules of mathematics had been established - the so called ZermeloFrankel axioms of set theory (ZFC). In the 1930's, Kurt Godelstunned the mathematical world with his incompleteness theoremsthat showed that any systematization such as ZFC isincomplete. I.e., there will always remain sentences that canneither be proved nor refuted within that systematization. Thisis normally referred to as the incompleteness phenomenom. Godel'soriginal examples of statements of unprovable and unrefutable inZFC were very far removed from the usual considerations ofmathematicians. Through a series of developments, starting withlater work of Godel and Cohen, various specialists in set theory,work of Friedman recognized by the NSF Alan T. Waterman Award in1984, and more recent work of Friedman, a body of such exampleshas been built up that are of increasing relevance to normalmathematical considerations. Recent work of Friedman along theselines gives new reasons for rethinking and extending the usualZFC axioms for mathematics. To test the broader impact of theresearch, Friedman actively seeks and obtains regular feedback onthe "naturalness" and "normality" of the various examples fromthe wider mathematical community. Friedman seeks to widen thisbroader impact.

奖项：DMS-0245349主要研究者：Harvey M.弗里德曼建议继续努力扩展不完全现象的范围。在先前NSF的支持下，弗里德曼发现了一种新的数学理论，该理论试图分析集合和它们的图像之间在多变量函数下的布尔关系。这种新的布尔关系理论试图分析以下形式的陈述：“对于某种类型的所有函数，存在某种类型的集合，使得给定的布尔关系在集合和它们在函数下的图像之间成立”。在先前NSF的支持下，弗里德曼发现了一个特别简单的布尔关系理论的“奇异”陈述，只有通过超越通常的数学公理才能证明。 弗里德曼已经考虑了所有6561个相同的简单形式的陈述，并表明所有这些陈述都可以用弱公理来证明或反驳，唯一的例外（直到对称性）是“奇异”陈述。 弗里德曼提出了从几个方面发展布尔关系理论，包括扩展6561个语句的集合，并将其转移到许多不同的数学环境中。到了世纪早期，数学的标准公理和规则已经建立起来--所谓的Zermelo-Frankel集合论公理（ZFC）。在20世纪30年代，库尔特·哥德尔用他的不完备性定理震惊了数学界，他的定理表明，任何像ZFC这样的系统化都是不完备的。即，总会有一些句子在系统化的体系中既不能被证明也不能被反驳。这通常被称为不完全现象。哥德尔在ZFC中关于不可证明和不可反驳的陈述的最初例子与数学家通常的考虑相去甚远。经过一系列的发展，从后来的工作哥德尔和科恩，各种专家在集合论，工作弗里德曼承认的国家科学基金会艾伦T。沃特曼奖在1984年，以及弗里德曼最近的工作，这样的例子已经建立了一个机构，越来越多的相关性正常的数学考虑。弗里德曼沿着这条路线的最新工作为重新思考和扩展数学中通常的ZFC公理提供了新的理由。为了测试这项研究的广泛影响，弗里德曼积极寻求并从更广泛的数学界获得有关各种例子的“自然性”和“正常性”的定期反馈。弗里德曼试图扩大这种更广泛的影响。