Research in the Foundations of Mathematics

奖项：DMS-0245349首席研究员：哈维·M·弗里德曼·弗里德曼提议继续努力，扩大不完整现象的范围。在先前NSF的支持下，弗里德曼发现了一种新的数学理论，该理论试图分析多变量函数下集合与其图像之间的布尔关系。这一新的布尔关系理论试图分析“对于某种类型的函数，存在某种类型的集合，使得给定的布尔关系在这些集合及其函数下的映象之间成立”这一形式的陈述。在先前NSF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，只有超越通常的数学公理才能证明这一陈述。弗里德曼考虑了所有相同简单形式的6561个陈述，并表明所有的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条语句的集合，并转移到许多不同的数学上下文。到20世纪初，数学的标准公理和规则已经建立-所谓的集合论零-弗兰克尔公理(Zfc)。在20世纪30年代的S中，库尔特·戈德勒以他的不完全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整的。也就是说，在这种系统化中，总是会有既不能证明也不能反驳的句子。这通常被称为不完全现象。戈德尔在ZFC中关于不可证明和不可反驳的陈述的重要例子，与数学家通常的考虑相去甚远。经过一系列的发展，从戈德尔和科恩后来的工作开始，集合论的各种专家，弗里德曼的工作在1984年被美国国家科学基金会艾伦·T·沃特曼奖认可，以及更近的弗里德曼的工作，已经建立了一系列这样的例子，与正常的数学考虑越来越相关。弗里德曼最近在这些方面的工作为重新思考和扩展通常的ZFC数学公理提供了新的理由。为了测试这项研究的广泛影响，弗里德曼积极地从更广泛的数学界寻求并获得对各种例子的“自然性”和“正规性”的定期反馈。弗里德曼试图扩大这一更广泛的影响。

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.