Research in the Foundations of Mathematics

奖项：dms -0245349首席研究员：Harvey M. FriedmanFriedman建议继续他的努力，扩大不完备现象的范围。在先前的国家科学基金会的支持下，弗里德曼发现了一个新的数学理论，该理论试图分析在多个变量函数下集合及其图像之间的布尔关系。这个新的布尔关系理论试图分析这样的命题：“对于所有的某一类函数，存在某一类集合，使得给定的布尔关系在这些集合及其在函数下的映像之间成立”。在先前的国家科学基金会的支持下，弗里德曼发现了布尔关系理论中一种特别简单形式的“奇异”陈述，这种陈述只能通过超越通常的数学公理来证明。弗里德曼考虑了6561个具有相同简单形式的陈述，并证明了所有陈述都可以用弱公理来证明或反驳，唯一的例外是“奇异”陈述（直到对称）。弗里德曼建议在几个方向上发展布尔关系理论，包括扩展6561个语句的集合，并转移到许多不同的数学背景。到20世纪早期，标准的数学公理和规则已经建立起来——即所谓的集论的泽尔梅洛-弗兰克尔公理（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。也就是说，总有一些句子在这个系统中既不能被证明也不能被反驳。这通常被称为不完备现象。哥德尔在zfc中关于不可证明和不可反驳的陈述的原始例子与数学家通常的考虑相去甚远。通过一系列的发展，从哥德尔和科恩后来的工作开始，集合论的各种专家，1984年获得国家科学基金会艾伦·t·沃特曼奖的弗里德曼的工作，以及弗里德曼最近的工作，已经建立了一个与常规数学考虑越来越相关的例子体。弗里德曼在这方面的最新工作为重新思考和扩展数学中常用的zfc公理提供了新的理由。为了测试研究的广泛影响，弗里德曼积极寻求并从更广泛的数学界获得关于各种例子的“自然性”和“正态性”的定期反馈。弗里德曼试图扩大这种更广泛的影响。

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.