Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the two-volume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of first-order logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent re-evaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higher-order reflection principles, and produced a stream of

Abstract: R. L. Moore (1882-1974) was one of the towering figures in American mathematics. This paper investigates the genesis of the special teaching method he adopted, now called the Moore Method. We examine the nine-year period that Moore spent at the University of Pennsylvania with a view toward singling out the essential ingredients in the method that he later perfected at the University of Texas. Moore’s enduring influence on a long list of distinguished mathematicians can be seen in the careers of his Penn colleague H. H. Mitchell, of his doctoral students J. R. Kline, G. H. Hallett, and A. M. Mullikin, and of some of Kline’s doctoral students. The Moore Method, named after the eminent topologist Robert Lee Moore (1884-1972), is perhaps the most well-known process in the world for training research mathematicians. In spite of its notoriety, however, there appear to be major misconceptions regarding its origin and development. For example, the editor of an MAA publication wrote, “R. L. Moore developed his approach to discovery learning from 1920 to 1969 at the University of Texas. ” [21, p. 6] On the contrary, we supply overwhelming evidence to support the contention that the pivotal period of development occurred between 1911 and 1920, when Moore was in Philadelphia and not in Austin. Overall we trace the evolution of the Moore Method through three distinct periods: